| L(s) = 1 | + 2.58·2-s + 0.785·3-s + 4.66·4-s + 2.02·6-s − 1.95·7-s + 6.87·8-s − 2.38·9-s + 4.90·11-s + 3.66·12-s − 3.73·13-s − 5.05·14-s + 8.41·16-s + 4.21·17-s − 6.15·18-s + 1.74·19-s − 1.53·21-s + 12.6·22-s − 23-s + 5.39·24-s − 9.64·26-s − 4.22·27-s − 9.13·28-s − 9.84·29-s − 4.31·31-s + 7.98·32-s + 3.85·33-s + 10.8·34-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 0.453·3-s + 2.33·4-s + 0.827·6-s − 0.740·7-s + 2.43·8-s − 0.794·9-s + 1.47·11-s + 1.05·12-s − 1.03·13-s − 1.35·14-s + 2.10·16-s + 1.02·17-s − 1.44·18-s + 0.399·19-s − 0.335·21-s + 2.69·22-s − 0.208·23-s + 1.10·24-s − 1.89·26-s − 0.813·27-s − 1.72·28-s − 1.82·29-s − 0.775·31-s + 1.41·32-s + 0.670·33-s + 1.86·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.392883951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.392883951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 3 | \( 1 - 0.785T + 3T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 11 | \( 1 - 4.90T + 11T^{2} \) |
| 13 | \( 1 + 3.73T + 13T^{2} \) |
| 17 | \( 1 - 4.21T + 17T^{2} \) |
| 19 | \( 1 - 1.74T + 19T^{2} \) |
| 29 | \( 1 + 9.84T + 29T^{2} \) |
| 31 | \( 1 + 4.31T + 31T^{2} \) |
| 37 | \( 1 + 5.25T + 37T^{2} \) |
| 41 | \( 1 - 6.89T + 41T^{2} \) |
| 43 | \( 1 + 6.95T + 43T^{2} \) |
| 47 | \( 1 - 3.53T + 47T^{2} \) |
| 53 | \( 1 + 8.27T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 3.44T + 67T^{2} \) |
| 71 | \( 1 - 3.53T + 71T^{2} \) |
| 73 | \( 1 + 5.93T + 73T^{2} \) |
| 79 | \( 1 + 0.224T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 0.368T + 97T^{2} \) |
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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
−11.23301511290371805093100461728, −9.886605225040789753225418152573, −9.102534749078379750748893156267, −7.65177050912936583016067053767, −6.85243138589513305453825529135, −5.91527597709411933343295960800, −5.18029166358178054799287502837, −3.78961515872589646175626690496, −3.33283722198859462048830363524, −2.08583493793738485406558410956,
2.08583493793738485406558410956, 3.33283722198859462048830363524, 3.78961515872589646175626690496, 5.18029166358178054799287502837, 5.91527597709411933343295960800, 6.85243138589513305453825529135, 7.65177050912936583016067053767, 9.102534749078379750748893156267, 9.886605225040789753225418152573, 11.23301511290371805093100461728
