| L(s) = 1 | + 1.87·2-s − 3-s + 1.51·4-s − 1.87·6-s + 3.55·7-s − 0.901·8-s + 9-s + 4.51·11-s − 1.51·12-s + 1.68·13-s + 6.66·14-s − 4.73·16-s − 3.85·17-s + 1.87·18-s + 6.13·19-s − 3.55·21-s + 8.47·22-s − 3.57·23-s + 0.901·24-s + 3.15·26-s − 27-s + 5.40·28-s − 4.07·29-s − 3.45·31-s − 7.07·32-s − 4.51·33-s − 7.23·34-s + ⋯ |
| L(s) = 1 | + 1.32·2-s − 0.577·3-s + 0.759·4-s − 0.765·6-s + 1.34·7-s − 0.318·8-s + 0.333·9-s + 1.36·11-s − 0.438·12-s + 0.466·13-s + 1.78·14-s − 1.18·16-s − 0.934·17-s + 0.442·18-s + 1.40·19-s − 0.775·21-s + 1.80·22-s − 0.746·23-s + 0.184·24-s + 0.618·26-s − 0.192·27-s + 1.02·28-s − 0.755·29-s − 0.620·31-s − 1.24·32-s − 0.786·33-s − 1.24·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.421513871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.421513871\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - 1.87T + 2T^{2} \) |
| 7 | \( 1 - 3.55T + 7T^{2} \) |
| 11 | \( 1 - 4.51T + 11T^{2} \) |
| 13 | \( 1 - 1.68T + 13T^{2} \) |
| 17 | \( 1 + 3.85T + 17T^{2} \) |
| 19 | \( 1 - 6.13T + 19T^{2} \) |
| 23 | \( 1 + 3.57T + 23T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 + 3.45T + 31T^{2} \) |
| 37 | \( 1 + 6.54T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 9.77T + 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 + 3.49T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 8.72T + 61T^{2} \) |
| 67 | \( 1 - 9.94T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 - 5.75T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 + 5.66T + 83T^{2} \) |
| 89 | \( 1 - 4.12T + 89T^{2} \) |
| 97 | \( 1 - 5.91T + 97T^{2} \) |
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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
−11.46960915569270822800061666832, −11.13280154753397099890578036562, −9.564934891286159083291362759923, −8.594541110671841934333318586304, −7.27652259600282119517514844443, −6.26107664583768325846801095831, −5.35641298281510186277249723508, −4.50031536908206476521831596291, −3.61598289342716126666884381596, −1.70470255037026491034768192931,
1.70470255037026491034768192931, 3.61598289342716126666884381596, 4.50031536908206476521831596291, 5.35641298281510186277249723508, 6.26107664583768325846801095831, 7.27652259600282119517514844443, 8.594541110671841934333318586304, 9.564934891286159083291362759923, 11.13280154753397099890578036562, 11.46960915569270822800061666832
