| L(s) = 1 | − 3·3-s − 16.8·5-s − 4.83·7-s + 9·9-s − 39.6·11-s − 13·13-s + 50.4·15-s − 4.33·17-s + 28.8·19-s + 14.4·21-s + 0.670·23-s + 158.·25-s − 27·27-s − 6·29-s + 274.·31-s + 118.·33-s + 81.3·35-s + 84.3·37-s + 39·39-s + 209.·41-s + 364.·43-s − 151.·45-s + 239.·47-s − 319.·49-s + 13.0·51-s − 415.·53-s + 667.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.50·5-s − 0.260·7-s + 0.333·9-s − 1.08·11-s − 0.277·13-s + 0.869·15-s − 0.0618·17-s + 0.348·19-s + 0.150·21-s + 0.00607·23-s + 1.26·25-s − 0.192·27-s − 0.0384·29-s + 1.59·31-s + 0.627·33-s + 0.392·35-s + 0.374·37-s + 0.160·39-s + 0.796·41-s + 1.29·43-s − 0.501·45-s + 0.742·47-s − 0.931·49-s + 0.0357·51-s − 1.07·53-s + 1.63·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7177120656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7177120656\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 16.8T + 125T^{2} \) |
| 7 | \( 1 + 4.83T + 343T^{2} \) |
| 11 | \( 1 + 39.6T + 1.33e3T^{2} \) |
| 17 | \( 1 + 4.33T + 4.91e3T^{2} \) |
| 19 | \( 1 - 28.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 0.670T + 1.21e4T^{2} \) |
| 29 | \( 1 + 6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 274.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 84.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 209.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 364.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 239.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 415.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 78T + 2.05e5T^{2} \) |
| 61 | \( 1 + 813.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 858.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 213.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 568.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 583.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 162.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 798.T + 9.12e5T^{2} \) |
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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
−11.25436728830919157413193400354, −10.57088422815317014673382692516, −9.442446914904792999766688582196, −8.062716069301654496009204325083, −7.57122280748334288161192085717, −6.39678236371818131417447254622, −5.08873577320498880915398998606, −4.14506193888781918586256226119, −2.84525773573596656463848730287, −0.58206734096837513085031327682,
0.58206734096837513085031327682, 2.84525773573596656463848730287, 4.14506193888781918586256226119, 5.08873577320498880915398998606, 6.39678236371818131417447254622, 7.57122280748334288161192085717, 8.062716069301654496009204325083, 9.442446914904792999766688582196, 10.57088422815317014673382692516, 11.25436728830919157413193400354
