| L(s) = 1 | + 2-s + 4-s − 3·5-s + 7-s + 8-s − 3·10-s + 4·13-s + 14-s + 16-s + 17-s − 7·19-s − 3·20-s + 4·25-s + 4·26-s + 28-s − 2·29-s + 7·31-s + 32-s + 34-s − 3·35-s + 5·37-s − 7·38-s − 3·40-s + 8·41-s − 43-s − 6·47-s − 6·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s − 0.948·10-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.60·19-s − 0.670·20-s + 4/5·25-s + 0.784·26-s + 0.188·28-s − 0.371·29-s + 1.25·31-s + 0.176·32-s + 0.171·34-s − 0.507·35-s + 0.821·37-s − 1.13·38-s − 0.474·40-s + 1.24·41-s − 0.152·43-s − 0.875·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.94902289137310, −14.80303923032859, −14.17238048596827, −13.53747857159935, −12.96734106961114, −12.57938446760542, −12.01634568470847, −11.45227133164793, −10.98165853681978, −10.82303395704017, −9.923525000370994, −9.219999958666348, −8.447135489709553, −8.057416793311385, −7.752470203677378, −6.920257399813082, −6.244868851004119, −6.024465180693491, −4.908775043750062, −4.523450272955792, −4.001139341671485, −3.436135331796734, −2.789256082421879, −1.875994303314791, −1.051554366721803, 0,
1.051554366721803, 1.875994303314791, 2.789256082421879, 3.436135331796734, 4.001139341671485, 4.523450272955792, 4.908775043750062, 6.024465180693491, 6.244868851004119, 6.920257399813082, 7.752470203677378, 8.057416793311385, 8.447135489709553, 9.219999958666348, 9.923525000370994, 10.82303395704017, 10.98165853681978, 11.45227133164793, 12.01634568470847, 12.57938446760542, 12.96734106961114, 13.53747857159935, 14.17238048596827, 14.80303923032859, 14.94902289137310
