| L(s) = 1 | − 3·3-s + 8·7-s + 9·9-s + 4·11-s + 6·13-s + 2·17-s − 16·19-s − 24·21-s + 60·23-s − 27·27-s − 142·29-s − 176·31-s − 12·33-s + 214·37-s − 18·39-s − 278·41-s + 68·43-s − 116·47-s − 279·49-s − 6·51-s + 350·53-s + 48·57-s + 684·59-s − 394·61-s + 72·63-s − 108·67-s − 180·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.431·7-s + 1/3·9-s + 0.109·11-s + 0.128·13-s + 0.0285·17-s − 0.193·19-s − 0.249·21-s + 0.543·23-s − 0.192·27-s − 0.909·29-s − 1.01·31-s − 0.0633·33-s + 0.950·37-s − 0.0739·39-s − 1.05·41-s + 0.241·43-s − 0.360·47-s − 0.813·49-s − 0.0164·51-s + 0.907·53-s + 0.111·57-s + 1.50·59-s − 0.826·61-s + 0.143·63-s − 0.196·67-s − 0.314·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 T + p^{3} T^{2} \) |
| 19 | \( 1 + 16 T + p^{3} T^{2} \) |
| 23 | \( 1 - 60 T + p^{3} T^{2} \) |
| 29 | \( 1 + 142 T + p^{3} T^{2} \) |
| 31 | \( 1 + 176 T + p^{3} T^{2} \) |
| 37 | \( 1 - 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 278 T + p^{3} T^{2} \) |
| 43 | \( 1 - 68 T + p^{3} T^{2} \) |
| 47 | \( 1 + 116 T + p^{3} T^{2} \) |
| 53 | \( 1 - 350 T + p^{3} T^{2} \) |
| 59 | \( 1 - 684 T + p^{3} T^{2} \) |
| 61 | \( 1 + 394 T + p^{3} T^{2} \) |
| 67 | \( 1 + 108 T + p^{3} T^{2} \) |
| 71 | \( 1 + 96 T + p^{3} T^{2} \) |
| 73 | \( 1 - 398 T + p^{3} T^{2} \) |
| 79 | \( 1 - 136 T + p^{3} T^{2} \) |
| 83 | \( 1 + 436 T + p^{3} T^{2} \) |
| 89 | \( 1 + 750 T + p^{3} T^{2} \) |
| 97 | \( 1 + 82 T + p^{3} T^{2} \) |
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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
−8.204014886930907303975792311501, −7.38467181238531736281837965032, −6.70159948407820517972584699592, −5.79647721090024484194261158103, −5.14863909018472010304420517546, −4.28658933905352103234580167564, −3.39871864216209988775539185267, −2.14898842688577162803221285387, −1.18253251751782500531315014379, 0,
1.18253251751782500531315014379, 2.14898842688577162803221285387, 3.39871864216209988775539185267, 4.28658933905352103234580167564, 5.14863909018472010304420517546, 5.79647721090024484194261158103, 6.70159948407820517972584699592, 7.38467181238531736281837965032, 8.204014886930907303975792311501
