| L(s) = 1 | + 28·2-s + 116·3-s + 272·4-s − 625·5-s + 3.24e3·6-s − 6.72e3·8-s − 6.22e3·9-s − 1.75e4·10-s − 2.55e4·11-s + 3.15e4·12-s + 4.23e4·13-s − 7.25e4·15-s − 3.27e5·16-s + 5.26e5·17-s − 1.74e5·18-s + 3.50e5·19-s − 1.70e5·20-s − 7.15e5·22-s − 6.21e5·23-s − 7.79e5·24-s + 3.90e5·25-s + 1.18e6·26-s − 3.00e6·27-s + 6.72e6·29-s − 2.03e6·30-s + 6.41e6·31-s − 5.72e6·32-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 0.826·3-s + 0.531·4-s − 0.447·5-s + 1.02·6-s − 0.580·8-s − 0.316·9-s − 0.553·10-s − 0.526·11-s + 0.439·12-s + 0.410·13-s − 0.369·15-s − 1.24·16-s + 1.52·17-s − 0.391·18-s + 0.616·19-s − 0.237·20-s − 0.651·22-s − 0.463·23-s − 0.479·24-s + 1/5·25-s + 0.508·26-s − 1.08·27-s + 1.76·29-s − 0.457·30-s + 1.24·31-s − 0.965·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.851182217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.851182217\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p^{4} T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 7 p^{2} T + p^{9} T^{2} \) |
| 3 | \( 1 - 116 T + p^{9} T^{2} \) |
| 11 | \( 1 + 25548 T + p^{9} T^{2} \) |
| 13 | \( 1 - 42306 T + p^{9} T^{2} \) |
| 17 | \( 1 - 526342 T + p^{9} T^{2} \) |
| 19 | \( 1 - 350060 T + p^{9} T^{2} \) |
| 23 | \( 1 + 621976 T + p^{9} T^{2} \) |
| 29 | \( 1 - 6720430 T + p^{9} T^{2} \) |
| 31 | \( 1 - 6412208 T + p^{9} T^{2} \) |
| 37 | \( 1 + 2317682 T + p^{9} T^{2} \) |
| 41 | \( 1 - 10224678 T + p^{9} T^{2} \) |
| 43 | \( 1 - 30114004 T + p^{9} T^{2} \) |
| 47 | \( 1 - 23644912 T + p^{9} T^{2} \) |
| 53 | \( 1 - 57292654 T + p^{9} T^{2} \) |
| 59 | \( 1 + 84934780 T + p^{9} T^{2} \) |
| 61 | \( 1 + 14677822 T + p^{9} T^{2} \) |
| 67 | \( 1 + 244557812 T + p^{9} T^{2} \) |
| 71 | \( 1 - 61901952 T + p^{9} T^{2} \) |
| 73 | \( 1 - 283763726 T + p^{9} T^{2} \) |
| 79 | \( 1 - 276107480 T + p^{9} T^{2} \) |
| 83 | \( 1 - 72995956 T + p^{9} T^{2} \) |
| 89 | \( 1 - 896368470 T + p^{9} T^{2} \) |
| 97 | \( 1 + 1205809578 T + p^{9} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.60059742670023461335473608538, −9.426438667594054551305704645620, −8.388209578007339359698255062721, −7.63663880485100099252242119864, −6.19229583255996524076856223336, −5.29758289969717486531659400351, −4.17798043047802162957861052153, −3.22404943495095347836639843266, −2.60580374080757144434534781033, −0.804447853208557462829340316134,
0.804447853208557462829340316134, 2.60580374080757144434534781033, 3.22404943495095347836639843266, 4.17798043047802162957861052153, 5.29758289969717486531659400351, 6.19229583255996524076856223336, 7.63663880485100099252242119864, 8.388209578007339359698255062721, 9.426438667594054551305704645620, 10.60059742670023461335473608538
