| L(s) = 1 | + 3·3-s − 10.6·5-s + 7·7-s + 9·9-s − 68.4·11-s − 23.0·13-s − 31.8·15-s − 62.2·17-s − 53.2·19-s + 21·21-s + 89.8·23-s − 11.9·25-s + 27·27-s + 43.2·29-s + 102.·31-s − 205.·33-s − 74.4·35-s + 302.·37-s − 69.1·39-s + 73.4·41-s + 377.·43-s − 95.6·45-s + 487.·47-s + 49·49-s − 186.·51-s − 467.·53-s + 727.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.951·5-s + 0.377·7-s + 0.333·9-s − 1.87·11-s − 0.492·13-s − 0.549·15-s − 0.887·17-s − 0.643·19-s + 0.218·21-s + 0.815·23-s − 0.0954·25-s + 0.192·27-s + 0.277·29-s + 0.595·31-s − 1.08·33-s − 0.359·35-s + 1.34·37-s − 0.284·39-s + 0.279·41-s + 1.33·43-s − 0.317·45-s + 1.51·47-s + 0.142·49-s − 0.512·51-s − 1.21·53-s + 1.78·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.463859274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.463859274\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 10.6T + 125T^{2} \) |
| 11 | \( 1 + 68.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 62.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 53.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 89.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 43.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 102.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 302.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 73.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 377.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 487.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 467.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 432.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 70.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 475.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 680.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 604.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 329.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 834.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 947.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 661.T + 9.12e5T^{2} \) |
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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
−9.066787988437718243284736060828, −8.285910133549708837761812942917, −7.71838590238666272119968768426, −7.14735309141460540240903010368, −5.87101019578253330557051672354, −4.75788579570462247633275537404, −4.22485262779167849496239875605, −2.90661288114489359232921896562, −2.28090561271363812353384884348, −0.56084243431055737046410897901,
0.56084243431055737046410897901, 2.28090561271363812353384884348, 2.90661288114489359232921896562, 4.22485262779167849496239875605, 4.75788579570462247633275537404, 5.87101019578253330557051672354, 7.14735309141460540240903010368, 7.71838590238666272119968768426, 8.285910133549708837761812942917, 9.066787988437718243284736060828
