L(s) = 1 | − 3-s + 2·5-s − 4·7-s − 2·9-s − 5·11-s + 13-s − 2·15-s + 2·17-s − 19-s + 4·21-s − 3·23-s − 25-s + 5·27-s − 10·29-s + 4·31-s + 5·33-s − 8·35-s − 7·37-s − 39-s + 3·41-s − 6·43-s − 4·45-s + 9·49-s − 2·51-s + 53-s − 10·55-s + 57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s − 2/3·9-s − 1.50·11-s + 0.277·13-s − 0.516·15-s + 0.485·17-s − 0.229·19-s + 0.872·21-s − 0.625·23-s − 1/5·25-s + 0.962·27-s − 1.85·29-s + 0.718·31-s + 0.870·33-s − 1.35·35-s − 1.15·37-s − 0.160·39-s + 0.468·41-s − 0.914·43-s − 0.596·45-s + 9/7·49-s − 0.280·51-s + 0.137·53-s − 1.34·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 107 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.47771256942928046550602438130, −10.01944994967627248707528054799, −9.095894430311072262951660930349, −7.944427580406038124448649385610, −6.69458978519928856970332081423, −5.83423828601274525106888297523, −5.33260552044036445015001522939, −3.50602259128846456078754707155, −2.38097312341385369240099608103, 0,
2.38097312341385369240099608103, 3.50602259128846456078754707155, 5.33260552044036445015001522939, 5.83423828601274525106888297523, 6.69458978519928856970332081423, 7.944427580406038124448649385610, 9.095894430311072262951660930349, 10.01944994967627248707528054799, 10.47771256942928046550602438130