| L(s) = 1 | + 15·4-s − 78·13-s + 161·16-s − 232·17-s − 284·23-s − 25·25-s + 440·29-s + 424·43-s + 430·49-s − 1.17e3·52-s + 756·53-s − 1.15e3·61-s + 1.45e3·64-s − 3.48e3·68-s − 2.80e3·79-s − 4.26e3·92-s − 375·100-s + 976·101-s + 3.22e3·103-s − 352·107-s − 3.70e3·113-s + 6.60e3·116-s − 938·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 15/8·4-s − 1.66·13-s + 2.51·16-s − 3.30·17-s − 2.57·23-s − 1/5·25-s + 2.81·29-s + 1.50·43-s + 1.25·49-s − 3.12·52-s + 1.95·53-s − 2.42·61-s + 2.84·64-s − 6.20·68-s − 3.98·79-s − 4.82·92-s − 3/8·100-s + 0.961·101-s + 3.08·103-s − 0.318·107-s − 3.08·113-s + 5.28·116-s − 0.704·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.212558012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.212558012\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 15 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 430 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 938 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 116 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13702 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 142 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 220 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 24518 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 99370 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 942 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 212 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 176670 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 378 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 235658 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 578 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 515090 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 696222 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 339790 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1400 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 516310 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1323502 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1743550 T^{2} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.44189118929549658830028838427, −10.29567441074794600299455520239, −9.906579168573679063576516748457, −9.121555995724561885713281764246, −8.745995672580483973594060831123, −8.267018715366674494435832076054, −7.69155392431097794769500303865, −7.27221762678945824466615799420, −6.99185304867128757931177720730, −6.39709069979057115816551818639, −6.19460933918012104404730873147, −5.71512841167115009464202635251, −4.85866050403971711122884184469, −4.29414935093721800997500160425, −4.06934060367558082668950497431, −2.79472382534234154415883114887, −2.52893176462434630763935359460, −2.25377715538115704735732866990, −1.56505552180723384502494730193, −0.38988158860387504009689970460,
0.38988158860387504009689970460, 1.56505552180723384502494730193, 2.25377715538115704735732866990, 2.52893176462434630763935359460, 2.79472382534234154415883114887, 4.06934060367558082668950497431, 4.29414935093721800997500160425, 4.85866050403971711122884184469, 5.71512841167115009464202635251, 6.19460933918012104404730873147, 6.39709069979057115816551818639, 6.99185304867128757931177720730, 7.27221762678945824466615799420, 7.69155392431097794769500303865, 8.267018715366674494435832076054, 8.745995672580483973594060831123, 9.121555995724561885713281764246, 9.906579168573679063576516748457, 10.29567441074794600299455520239, 10.44189118929549658830028838427
