| L(s) = 1 | + 2-s + 4-s + 5-s + 2·7-s + 8-s + 10-s + 4·11-s + 4·13-s + 2·14-s + 16-s − 17-s − 4·19-s + 20-s + 4·22-s − 8·23-s + 25-s + 4·26-s + 2·28-s − 2·29-s + 4·31-s + 32-s − 34-s + 2·35-s + 6·37-s − 4·38-s + 40-s − 8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s + 0.353·8-s + 0.316·10-s + 1.20·11-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.917·19-s + 0.223·20-s + 0.852·22-s − 1.66·23-s + 1/5·25-s + 0.784·26-s + 0.377·28-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.171·34-s + 0.338·35-s + 0.986·37-s − 0.648·38-s + 0.158·40-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.384298443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.384298443\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−9.494105974217632028628710487901, −8.517159476126515540923697383200, −7.975459727897265896880130970982, −6.62658089655629739547596572135, −6.28351765143151982678760067799, −5.33818292420531778964325455979, −4.27379736541190030115205025603, −3.73946400941938659966316638969, −2.27240551471290234586838368449, −1.38178709741572518513115318201,
1.38178709741572518513115318201, 2.27240551471290234586838368449, 3.73946400941938659966316638969, 4.27379736541190030115205025603, 5.33818292420531778964325455979, 6.28351765143151982678760067799, 6.62658089655629739547596572135, 7.975459727897265896880130970982, 8.517159476126515540923697383200, 9.494105974217632028628710487901
