L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s + 13-s + 4·14-s + 16-s − 2·17-s + 4·19-s + 8·23-s − 26-s − 4·28-s − 2·29-s − 8·31-s − 32-s + 2·34-s − 2·37-s − 4·38-s + 6·41-s − 12·43-s − 8·46-s + 9·49-s + 52-s + 10·53-s + 4·56-s + 2·58-s − 10·61-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 0.196·26-s − 0.755·28-s − 0.371·29-s − 1.43·31-s − 0.176·32-s + 0.342·34-s − 0.328·37-s − 0.648·38-s + 0.937·41-s − 1.82·43-s − 1.17·46-s + 9/7·49-s + 0.138·52-s + 1.37·53-s + 0.534·56-s + 0.262·58-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.66989657632459004812788027657, −6.99710497747098715630132652164, −6.59349388023917473502623632939, −5.73084052598874245562033811102, −5.02052022396818729106165437895, −3.70985819173759386781821472300, −3.23676801370547150675630240730, −2.33588772540446155513557717058, −1.11529347123600815773378494836, 0,
1.11529347123600815773378494836, 2.33588772540446155513557717058, 3.23676801370547150675630240730, 3.70985819173759386781821472300, 5.02052022396818729106165437895, 5.73084052598874245562033811102, 6.59349388023917473502623632939, 6.99710497747098715630132652164, 7.66989657632459004812788027657