| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 2·11-s + 16-s − 2·17-s − 4·19-s + 20-s + 2·22-s + 4·23-s + 25-s + 4·29-s − 31-s − 32-s + 2·34-s − 8·37-s + 4·38-s − 40-s − 6·41-s + 2·43-s − 2·44-s − 4·46-s − 7·49-s − 50-s − 8·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.603·11-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.223·20-s + 0.426·22-s + 0.834·23-s + 1/5·25-s + 0.742·29-s − 0.179·31-s − 0.176·32-s + 0.342·34-s − 1.31·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-s + 0.304·43-s − 0.301·44-s − 0.589·46-s − 49-s − 0.141·50-s − 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.509196251671545492180031763812, −7.82324426194698016701524671113, −6.85566084303057349352756596641, −6.38928498895932353355711112534, −5.36013908964288749051473827734, −4.62364450878552237809302992665, −3.35246501897724533562865353410, −2.44569227539742541309238526143, −1.50002343442867035876074318730, 0,
1.50002343442867035876074318730, 2.44569227539742541309238526143, 3.35246501897724533562865353410, 4.62364450878552237809302992665, 5.36013908964288749051473827734, 6.38928498895932353355711112534, 6.85566084303057349352756596641, 7.82324426194698016701524671113, 8.509196251671545492180031763812
