| L(s) = 1 | − 7-s − 11-s + 13-s − 3·17-s + 4·19-s − 2·23-s + 29-s + 6·31-s − 2·37-s + 10·41-s − 9·47-s + 49-s − 14·53-s + 6·59-s − 4·61-s + 10·67-s − 16·71-s − 10·73-s + 77-s + 11·79-s − 4·83-s − 12·89-s − 91-s + 19·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.301·11-s + 0.277·13-s − 0.727·17-s + 0.917·19-s − 0.417·23-s + 0.185·29-s + 1.07·31-s − 0.328·37-s + 1.56·41-s − 1.31·47-s + 1/7·49-s − 1.92·53-s + 0.781·59-s − 0.512·61-s + 1.22·67-s − 1.89·71-s − 1.17·73-s + 0.113·77-s + 1.23·79-s − 0.439·83-s − 1.27·89-s − 0.104·91-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−15.80448578107157, −15.19422148426962, −14.39982990296833, −14.12320397948025, −13.31197371472327, −13.13304738176876, −12.38208504900600, −11.87121346443010, −11.27265065642240, −10.80287380618937, −10.09770372995495, −9.641387510324116, −9.108060505060298, −8.397699159918275, −7.906425708928285, −7.275383389790605, −6.584611416032819, −6.111723018136118, −5.449813295810962, −4.695129442052681, −4.185607255865452, −3.276882779070461, −2.808182026152841, −1.922142467276229, −1.027484862628749, 0,
1.027484862628749, 1.922142467276229, 2.808182026152841, 3.276882779070461, 4.185607255865452, 4.695129442052681, 5.449813295810962, 6.111723018136118, 6.584611416032819, 7.275383389790605, 7.906425708928285, 8.397699159918275, 9.108060505060298, 9.641387510324116, 10.09770372995495, 10.80287380618937, 11.27265065642240, 11.87121346443010, 12.38208504900600, 13.13304738176876, 13.31197371472327, 14.12320397948025, 14.39982990296833, 15.19422148426962, 15.80448578107157
