| L(s) = 1 | − 3-s − 2·4-s + 3.87·7-s + 9-s + 2·12-s − 3.87·13-s + 4·16-s − 7.74·17-s − 3.87·19-s − 3.87·21-s − 6·23-s − 27-s − 7.74·28-s − 7.74·29-s − 5·31-s − 2·36-s − 2·37-s + 3.87·39-s + 7.74·41-s + 3.87·43-s + 12·47-s − 4·48-s + 8.00·49-s + 7.74·51-s + 7.74·52-s − 6·53-s + 3.87·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1.46·7-s + 0.333·9-s + 0.577·12-s − 1.07·13-s + 16-s − 1.87·17-s − 0.888·19-s − 0.845·21-s − 1.25·23-s − 0.192·27-s − 1.46·28-s − 1.43·29-s − 0.898·31-s − 0.333·36-s − 0.328·37-s + 0.620·39-s + 1.20·41-s + 0.590·43-s + 1.75·47-s − 0.577·48-s + 1.14·49-s + 1.08·51-s + 1.07·52-s − 0.824·53-s + 0.512·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6926607207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6926607207\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 - 3.87T + 7T^{2} \) |
| 13 | \( 1 + 3.87T + 13T^{2} \) |
| 17 | \( 1 + 7.74T + 17T^{2} \) |
| 19 | \( 1 + 3.87T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 7.74T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 7.74T + 41T^{2} \) |
| 43 | \( 1 - 3.87T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 7.74T + 79T^{2} \) |
| 83 | \( 1 - 7.74T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−7.72443344173035650305227269758, −7.22714703094437013209002528885, −6.18982878107600860683823502873, −5.54089644552202212887165483081, −4.86958913500821922605921390980, −4.33590188077352888116422970479, −3.92040943924689634108825373133, −2.31340386250551021891451755849, −1.79190390893820911809888189928, −0.40790868698431077908871908363,
0.40790868698431077908871908363, 1.79190390893820911809888189928, 2.31340386250551021891451755849, 3.92040943924689634108825373133, 4.33590188077352888116422970479, 4.86958913500821922605921390980, 5.54089644552202212887165483081, 6.18982878107600860683823502873, 7.22714703094437013209002528885, 7.72443344173035650305227269758
