| L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 2·7-s − 2·10-s − 11-s + 4·13-s − 4·14-s − 4·16-s + 2·17-s − 2·20-s − 2·22-s + 23-s − 4·25-s + 8·26-s − 4·28-s + 7·31-s − 8·32-s + 4·34-s + 2·35-s + 3·37-s + 8·41-s − 6·43-s − 2·44-s + 2·46-s − 8·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 0.755·7-s − 0.632·10-s − 0.301·11-s + 1.10·13-s − 1.06·14-s − 16-s + 0.485·17-s − 0.447·20-s − 0.426·22-s + 0.208·23-s − 4/5·25-s + 1.56·26-s − 0.755·28-s + 1.25·31-s − 1.41·32-s + 0.685·34-s + 0.338·35-s + 0.493·37-s + 1.24·41-s − 0.914·43-s − 0.301·44-s + 0.294·46-s − 1.16·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.684496332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.684496332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.68040274047176229306789941909, −13.05532878430427737685314941779, −12.07025537765623530596257929043, −11.14136051051457299622132768480, −9.701226498296212058900989539920, −8.228854358433494441874279371750, −6.66503987304844816686807720890, −5.66407660024727617127120629681, −4.19085115174357929506261869806, −3.09000916592887094247460325417,
3.09000916592887094247460325417, 4.19085115174357929506261869806, 5.66407660024727617127120629681, 6.66503987304844816686807720890, 8.228854358433494441874279371750, 9.701226498296212058900989539920, 11.14136051051457299622132768480, 12.07025537765623530596257929043, 13.05532878430427737685314941779, 13.68040274047176229306789941909
