| L(s) = 1 | − 2-s + 4-s − 3·7-s − 8-s − 2·11-s + 6·13-s + 3·14-s + 16-s − 2·17-s + 19-s + 2·22-s − 3·23-s − 6·26-s − 3·28-s − 6·29-s + 2·31-s − 32-s + 2·34-s + 2·37-s − 38-s + 5·41-s + 8·43-s − 2·44-s + 3·46-s − 8·47-s + 2·49-s + 6·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.13·7-s − 0.353·8-s − 0.603·11-s + 1.66·13-s + 0.801·14-s + 1/4·16-s − 0.485·17-s + 0.229·19-s + 0.426·22-s − 0.625·23-s − 1.17·26-s − 0.566·28-s − 1.11·29-s + 0.359·31-s − 0.176·32-s + 0.342·34-s + 0.328·37-s − 0.162·38-s + 0.780·41-s + 1.21·43-s − 0.301·44-s + 0.442·46-s − 1.16·47-s + 2/7·49-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.15813568422644, −13.83029351572967, −13.19047783319527, −12.77591324770374, −12.52883275026744, −11.53384735490169, −11.28672996301354, −10.76737876774603, −10.26828458725348, −9.699420549364091, −9.267953738709308, −8.853593705684484, −8.168315713042959, −7.793239882123756, −7.190375806556391, −6.423222421574095, −6.209449615628719, −5.711833286304847, −4.928029237997891, −4.053881443787461, −3.589324142754703, −3.004632121254767, −2.334604602470011, −1.590430932490741, −0.7734915648694529, 0,
0.7734915648694529, 1.590430932490741, 2.334604602470011, 3.004632121254767, 3.589324142754703, 4.053881443787461, 4.928029237997891, 5.711833286304847, 6.209449615628719, 6.423222421574095, 7.190375806556391, 7.793239882123756, 8.168315713042959, 8.853593705684484, 9.267953738709308, 9.699420549364091, 10.26828458725348, 10.76737876774603, 11.28672996301354, 11.53384735490169, 12.52883275026744, 12.77591324770374, 13.19047783319527, 13.83029351572967, 14.15813568422644
