| L(s) = 1 | + 3·3-s + 12.9·5-s + 1.76·7-s + 9·9-s + 25.1·11-s + 13·13-s + 38.8·15-s − 62.6·17-s + 139.·19-s + 5.29·21-s − 56.6·23-s + 42.9·25-s + 27·27-s + 75.4·29-s − 71.2·31-s + 75.5·33-s + 22.8·35-s + 55.7·37-s + 39·39-s − 40.0·41-s + 14.7·43-s + 116.·45-s + 531.·47-s − 339.·49-s − 187.·51-s + 368.·53-s + 326.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.15·5-s + 0.0952·7-s + 0.333·9-s + 0.690·11-s + 0.277·13-s + 0.669·15-s − 0.893·17-s + 1.68·19-s + 0.0550·21-s − 0.513·23-s + 0.343·25-s + 0.192·27-s + 0.482·29-s − 0.413·31-s + 0.398·33-s + 0.110·35-s + 0.247·37-s + 0.160·39-s − 0.152·41-s + 0.0524·43-s + 0.386·45-s + 1.64·47-s − 0.990·49-s − 0.515·51-s + 0.953·53-s + 0.800·55-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
| \(L(2)\) |
\(\approx\) |
\(2.978766682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.978766682\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 12.9T + 125T^{2} \) |
| 7 | \( 1 - 1.76T + 343T^{2} \) |
| 11 | \( 1 - 25.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 62.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 56.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 75.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 71.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 55.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 40.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 14.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 531.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 368.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 165.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 145.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 901.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 345.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 292.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 722.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 565.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 275.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.82e3T + 9.12e5T^{2} \) |
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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
−11.20773849669217654583099184191, −10.04958611653186648911677439601, −9.405473028016121317522936551687, −8.606289509820526759436802645793, −7.36384634085107473617108983116, −6.33426764879916427070692880253, −5.31523515560876524689158029908, −3.92940301469109970451363992625, −2.54734859652549177684038742721, −1.34536899869049717706855553257,
1.34536899869049717706855553257, 2.54734859652549177684038742721, 3.92940301469109970451363992625, 5.31523515560876524689158029908, 6.33426764879916427070692880253, 7.36384634085107473617108983116, 8.606289509820526759436802645793, 9.405473028016121317522936551687, 10.04958611653186648911677439601, 11.20773849669217654583099184191
