| L(s) = 1 | − 3.15·3-s + 4.03·5-s + 6.93·9-s + 2.72·11-s − 3.56·13-s − 12.7·15-s − 1.69·17-s + 4.34·19-s − 1.78·23-s + 11.2·25-s − 12.4·27-s + 0.407·29-s − 6.58·31-s − 8.57·33-s − 9.98·37-s + 11.2·39-s − 7.86·41-s − 6.09·43-s + 27.9·45-s − 1.71·47-s + 5.33·51-s − 5.92·53-s + 10.9·55-s − 13.7·57-s − 7.68·59-s − 4.52·61-s − 14.3·65-s + ⋯ |
| L(s) = 1 | − 1.81·3-s + 1.80·5-s + 2.31·9-s + 0.820·11-s − 0.990·13-s − 3.28·15-s − 0.410·17-s + 0.997·19-s − 0.371·23-s + 2.25·25-s − 2.38·27-s + 0.0757·29-s − 1.18·31-s − 1.49·33-s − 1.64·37-s + 1.80·39-s − 1.22·41-s − 0.930·43-s + 4.16·45-s − 0.249·47-s + 0.746·51-s − 0.813·53-s + 1.48·55-s − 1.81·57-s − 1.00·59-s − 0.578·61-s − 1.78·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.15T + 3T^{2} \) |
| 5 | \( 1 - 4.03T + 5T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 + 1.69T + 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 + 1.78T + 23T^{2} \) |
| 29 | \( 1 - 0.407T + 29T^{2} \) |
| 31 | \( 1 + 6.58T + 31T^{2} \) |
| 37 | \( 1 + 9.98T + 37T^{2} \) |
| 41 | \( 1 + 7.86T + 41T^{2} \) |
| 43 | \( 1 + 6.09T + 43T^{2} \) |
| 47 | \( 1 + 1.71T + 47T^{2} \) |
| 53 | \( 1 + 5.92T + 53T^{2} \) |
| 59 | \( 1 + 7.68T + 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 + 4.16T + 67T^{2} \) |
| 71 | \( 1 - 7.93T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 0.285T + 79T^{2} \) |
| 83 | \( 1 + 9.36T + 83T^{2} \) |
| 89 | \( 1 + 8.94T + 89T^{2} \) |
| 97 | \( 1 - 7.00T + 97T^{2} \) |
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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
−7.34697071228573366158214847756, −6.83202218198601839622926566060, −6.27957895728139152044223588710, −5.60928824325474844177852316919, −5.14806175125579965596516318177, −4.55773190217699686082284657736, −3.26798431648436724480585152523, −1.87663914197817783895723822927, −1.42471268024790358241123998296, 0,
1.42471268024790358241123998296, 1.87663914197817783895723822927, 3.26798431648436724480585152523, 4.55773190217699686082284657736, 5.14806175125579965596516318177, 5.60928824325474844177852316919, 6.27957895728139152044223588710, 6.83202218198601839622926566060, 7.34697071228573366158214847756
