| L(s) = 1 | − 32·2-s − 162·3-s + 768·4-s + 1.25e3·5-s + 5.18e3·6-s − 4.80e3·7-s − 1.63e4·8-s + 1.96e4·9-s − 4.00e4·10-s + 1.49e4·11-s − 1.24e5·12-s − 1.14e5·13-s + 1.53e5·14-s − 2.02e5·15-s + 3.27e5·16-s − 2.64e5·17-s − 6.29e5·18-s + 4.73e5·19-s + 9.60e5·20-s + 7.77e5·21-s − 4.79e5·22-s − 4.18e5·23-s + 2.65e6·24-s + 1.17e6·25-s + 3.65e6·26-s − 2.12e6·27-s − 3.68e6·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s − 1.26·10-s + 0.308·11-s − 1.73·12-s − 1.10·13-s + 1.06·14-s − 1.03·15-s + 5/4·16-s − 0.768·17-s − 1.41·18-s + 0.834·19-s + 1.34·20-s + 0.872·21-s − 0.436·22-s − 0.311·23-s + 1.63·24-s + 3/5·25-s + 1.56·26-s − 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.8191199314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8191199314\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 14992 T + 2626147862 T^{2} - 14992 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 114156 T + 19638537374 T^{2} + 114156 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 264652 T + 189771312806 T^{2} + 264652 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 473856 T + 692926522598 T^{2} - 473856 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 418464 T + 1778581212046 T^{2} + 418464 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2253084 T + 19617542791198 T^{2} - 2253084 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4729768 T + 19071971009454 T^{2} - 4729768 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 664540 T + 260033346458670 T^{2} - 664540 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6749348 T + 587044452067094 T^{2} - 6749348 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4666568 T + 965821725862278 T^{2} - 4666568 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 21343216 T + 2352023454524798 T^{2} + 21343216 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 87480380 T + 7118653127703470 T^{2} + 87480380 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 160839272 T + 22832871542132678 T^{2} + 160839272 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 59352436 T - 2965994956048194 T^{2} + 59352436 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 253373496 T + 34345385799647798 T^{2} + 253373496 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 137783848 T + 85805037653747582 T^{2} + 137783848 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 470419028 T + 119686157381912022 T^{2} + 470419028 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 393396496 T + 2882897657571522 p T^{2} + 393396496 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 185061704 T + 272152951510785014 T^{2} - 185061704 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1046634436 T + 945004964934187958 T^{2} - 1046634436 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 945627148 T + 1743729683270266566 T^{2} - 945627148 p^{9} T^{3} + p^{18} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.63518221239546457132281143781, −10.48396800341462643240082181448, −9.831125326726487850847914956458, −9.765230029469425877906474376998, −8.987319435442312229699262965373, −8.932968801671378637450687621455, −7.75349063290440508577082639016, −7.60903768874149709638340387099, −6.82636193286142531627012714213, −6.53662743486226062523607534662, −5.99097848243843047469247038580, −5.70191549239175422672557935175, −4.69050801955348985869055294734, −4.51206647616565887676718795604, −3.08957890421972661095650746296, −2.91642636569226072807221425052, −1.86222428805829233767969616849, −1.62596639960405901963309539777, −0.68968736876128142130513551296, −0.39339298724814378779653464874,
0.39339298724814378779653464874, 0.68968736876128142130513551296, 1.62596639960405901963309539777, 1.86222428805829233767969616849, 2.91642636569226072807221425052, 3.08957890421972661095650746296, 4.51206647616565887676718795604, 4.69050801955348985869055294734, 5.70191549239175422672557935175, 5.99097848243843047469247038580, 6.53662743486226062523607534662, 6.82636193286142531627012714213, 7.60903768874149709638340387099, 7.75349063290440508577082639016, 8.932968801671378637450687621455, 8.987319435442312229699262965373, 9.765230029469425877906474376998, 9.831125326726487850847914956458, 10.48396800341462643240082181448, 10.63518221239546457132281143781
