| L(s) = 1 | − 3·9-s + 4·11-s + 6·17-s − 4·19-s − 2·29-s + 4·31-s − 6·37-s + 6·41-s − 8·43-s − 7·49-s − 2·53-s − 4·59-s − 10·61-s + 12·67-s + 4·71-s + 14·73-s − 16·79-s + 9·81-s + 12·83-s − 2·89-s − 2·97-s − 12·99-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 9-s + 1.20·11-s + 1.45·17-s − 0.917·19-s − 0.371·29-s + 0.718·31-s − 0.986·37-s + 0.937·41-s − 1.21·43-s − 49-s − 0.274·53-s − 0.520·59-s − 1.28·61-s + 1.46·67-s + 0.474·71-s + 1.63·73-s − 1.80·79-s + 81-s + 1.31·83-s − 0.211·89-s − 0.203·97-s − 1.20·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−15.15579788608361, −14.62223803074872, −14.18944666583527, −13.94192832298432, −13.13234398775342, −12.49384727303497, −12.07203193037243, −11.63330312464992, −11.02533750992105, −10.57256699147007, −9.736796565717181, −9.471779093940182, −8.715240076355853, −8.281907312920932, −7.776602901706109, −6.978378664665834, −6.365326689662462, −6.000267453130637, −5.245272514995404, −4.702843194268780, −3.781639310741808, −3.421201468979213, −2.648393309895716, −1.785554155708471, −1.048022480492480, 0,
1.048022480492480, 1.785554155708471, 2.648393309895716, 3.421201468979213, 3.781639310741808, 4.702843194268780, 5.245272514995404, 6.000267453130637, 6.365326689662462, 6.978378664665834, 7.776602901706109, 8.281907312920932, 8.715240076355853, 9.471779093940182, 9.736796565717181, 10.57256699147007, 11.02533750992105, 11.63330312464992, 12.07203193037243, 12.49384727303497, 13.13234398775342, 13.94192832298432, 14.18944666583527, 14.62223803074872, 15.15579788608361
