| L(s) = 1 | − 3·3-s + 5·5-s − 8·7-s + 9·9-s − 4·11-s + 6·13-s − 15·15-s − 2·17-s + 16·19-s + 24·21-s − 60·23-s + 25·25-s − 27·27-s + 142·29-s − 176·31-s + 12·33-s − 40·35-s + 214·37-s − 18·39-s − 278·41-s + 68·43-s + 45·45-s + 116·47-s − 279·49-s + 6·51-s + 350·53-s − 20·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.431·7-s + 1/3·9-s − 0.109·11-s + 0.128·13-s − 0.258·15-s − 0.0285·17-s + 0.193·19-s + 0.249·21-s − 0.543·23-s + 1/5·25-s − 0.192·27-s + 0.909·29-s − 1.01·31-s + 0.0633·33-s − 0.193·35-s + 0.950·37-s − 0.0739·39-s − 1.05·41-s + 0.241·43-s + 0.149·45-s + 0.360·47-s − 0.813·49-s + 0.0164·51-s + 0.907·53-s − 0.0490·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 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
−9.393031140566850928998629532300, −8.436197582349889265637397950414, −7.41323303448208328912061348618, −6.52576901237690635692444334310, −5.82655448275194688776318874314, −4.94372015506654880974673514019, −3.86090928985552313717829994001, −2.66047918213868617164136266601, −1.36664925703579617680327156457, 0,
1.36664925703579617680327156457, 2.66047918213868617164136266601, 3.86090928985552313717829994001, 4.94372015506654880974673514019, 5.82655448275194688776318874314, 6.52576901237690635692444334310, 7.41323303448208328912061348618, 8.436197582349889265637397950414, 9.393031140566850928998629532300
