| L(s) = 1 | + 2-s − 0.703·3-s + 4-s + 0.296·5-s − 0.703·6-s + 7-s + 8-s − 2.50·9-s + 0.296·10-s − 5.98·11-s − 0.703·12-s + 2.50·13-s + 14-s − 0.208·15-s + 16-s + 3.27·17-s − 2.50·18-s + 6.68·19-s + 0.296·20-s − 0.703·21-s − 5.98·22-s − 0.703·24-s − 4.91·25-s + 2.50·26-s + 3.87·27-s + 28-s − 2.27·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.406·3-s + 0.5·4-s + 0.132·5-s − 0.287·6-s + 0.377·7-s + 0.353·8-s − 0.835·9-s + 0.0937·10-s − 1.80·11-s − 0.203·12-s + 0.694·13-s + 0.267·14-s − 0.0538·15-s + 0.250·16-s + 0.795·17-s − 0.590·18-s + 1.53·19-s + 0.0663·20-s − 0.153·21-s − 1.27·22-s − 0.143·24-s − 0.982·25-s + 0.491·26-s + 0.745·27-s + 0.188·28-s − 0.423·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.542654607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.542654607\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 0.703T + 3T^{2} \) |
| 5 | \( 1 - 0.296T + 5T^{2} \) |
| 11 | \( 1 + 5.98T + 11T^{2} \) |
| 13 | \( 1 - 2.50T + 13T^{2} \) |
| 17 | \( 1 - 3.27T + 17T^{2} \) |
| 19 | \( 1 - 6.68T + 19T^{2} \) |
| 29 | \( 1 + 2.27T + 29T^{2} \) |
| 31 | \( 1 - 2.80T + 31T^{2} \) |
| 37 | \( 1 - 4.80T + 37T^{2} \) |
| 41 | \( 1 + 3.09T + 41T^{2} \) |
| 43 | \( 1 + 4.09T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 0.279T + 53T^{2} \) |
| 59 | \( 1 + 2.47T + 59T^{2} \) |
| 61 | \( 1 - 3.72T + 61T^{2} \) |
| 67 | \( 1 + 9.48T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 - 7.50T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−7.66917546482265266767126660505, −7.39658209030878763716046124245, −6.05326434688217496404326371247, −5.78194928350821216228879407464, −5.20062220781534881562842184274, −4.55605466392318645901764905221, −3.37772009050970962399322605861, −2.92559924945595629592978060692, −1.95208074512707627757797136947, −0.72397068007173777457049751413,
0.72397068007173777457049751413, 1.95208074512707627757797136947, 2.92559924945595629592978060692, 3.37772009050970962399322605861, 4.55605466392318645901764905221, 5.20062220781534881562842184274, 5.78194928350821216228879407464, 6.05326434688217496404326371247, 7.39658209030878763716046124245, 7.66917546482265266767126660505
