| L(s) = 1 | − 3·3-s + 32·7-s + 9·9-s − 36·11-s + 10·13-s + 78·17-s − 140·19-s − 96·21-s − 192·23-s − 27·27-s + 6·29-s + 16·31-s + 108·33-s + 34·37-s − 30·39-s − 390·41-s − 52·43-s + 408·47-s + 681·49-s − 234·51-s + 114·53-s + 420·57-s − 516·59-s − 58·61-s + 288·63-s − 892·67-s + 576·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.72·7-s + 1/3·9-s − 0.986·11-s + 0.213·13-s + 1.11·17-s − 1.69·19-s − 0.997·21-s − 1.74·23-s − 0.192·27-s + 0.0384·29-s + 0.0926·31-s + 0.569·33-s + 0.151·37-s − 0.123·39-s − 1.48·41-s − 0.184·43-s + 1.26·47-s + 1.98·49-s − 0.642·51-s + 0.295·53-s + 0.975·57-s − 1.13·59-s − 0.121·61-s + 0.575·63-s − 1.62·67-s + 1.00·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 140 T + p^{3} T^{2} \) |
| 23 | \( 1 + 192 T + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 - 16 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 390 T + p^{3} T^{2} \) |
| 43 | \( 1 + 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 408 T + p^{3} T^{2} \) |
| 53 | \( 1 - 114 T + p^{3} T^{2} \) |
| 59 | \( 1 + 516 T + p^{3} T^{2} \) |
| 61 | \( 1 + 58 T + p^{3} T^{2} \) |
| 67 | \( 1 + 892 T + p^{3} T^{2} \) |
| 71 | \( 1 - 120 T + p^{3} T^{2} \) |
| 73 | \( 1 - 646 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1168 T + p^{3} T^{2} \) |
| 83 | \( 1 + 732 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1590 T + p^{3} T^{2} \) |
| 97 | \( 1 + 2 p T + p^{3} T^{2} \) |
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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
−8.735229703157067682633599621954, −8.045205815811440097860446477827, −7.56239864962174165790575892292, −6.28499740097241640823993570310, −5.49479052150955976739072968344, −4.76123073111374433511542300960, −3.92748470922967557762775377781, −2.32444286707874308619545067967, −1.43649291851529844981664610270, 0,
1.43649291851529844981664610270, 2.32444286707874308619545067967, 3.92748470922967557762775377781, 4.76123073111374433511542300960, 5.49479052150955976739072968344, 6.28499740097241640823993570310, 7.56239864962174165790575892292, 8.045205815811440097860446477827, 8.735229703157067682633599621954
