| L(s) = 1 | + 2.21·2-s − 3-s + 2.90·4-s − 2.21·6-s − 0.903·7-s + 2·8-s + 9-s − 4.59·11-s − 2.90·12-s + 4.49·13-s − 2·14-s − 1.37·16-s − 6.73·17-s + 2.21·18-s − 3.62·19-s + 0.903·21-s − 10.1·22-s − 8.39·23-s − 2·24-s + 9.95·26-s − 27-s − 2.62·28-s − 1.68·29-s − 31-s − 7.05·32-s + 4.59·33-s − 14.9·34-s + ⋯ |
| L(s) = 1 | + 1.56·2-s − 0.577·3-s + 1.45·4-s − 0.903·6-s − 0.341·7-s + 0.707·8-s + 0.333·9-s − 1.38·11-s − 0.838·12-s + 1.24·13-s − 0.534·14-s − 0.344·16-s − 1.63·17-s + 0.521·18-s − 0.830·19-s + 0.197·21-s − 2.16·22-s − 1.75·23-s − 0.408·24-s + 1.95·26-s − 0.192·27-s − 0.495·28-s − 0.313·29-s − 0.179·31-s − 1.24·32-s + 0.799·33-s − 2.55·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.21T + 2T^{2} \) |
| 7 | \( 1 + 0.903T + 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 - 4.49T + 13T^{2} \) |
| 17 | \( 1 + 6.73T + 17T^{2} \) |
| 19 | \( 1 + 3.62T + 19T^{2} \) |
| 23 | \( 1 + 8.39T + 23T^{2} \) |
| 29 | \( 1 + 1.68T + 29T^{2} \) |
| 37 | \( 1 - 8.68T + 37T^{2} \) |
| 41 | \( 1 - 4.02T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 1.18T + 47T^{2} \) |
| 53 | \( 1 - 0.873T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 8.42T + 73T^{2} \) |
| 79 | \( 1 + 1.11T + 79T^{2} \) |
| 83 | \( 1 + 3.96T + 83T^{2} \) |
| 89 | \( 1 + 9.50T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−8.483474537311554242253608351183, −7.61432299525255518283798725526, −6.57514151104056134418911092190, −6.04092095374430536601514260712, −5.55753287549069786827950034516, −4.35177636749552801250470686707, −4.15498943305707178695911210096, −2.87051902058325295843232817121, −2.04810188065349367155872801904, 0,
2.04810188065349367155872801904, 2.87051902058325295843232817121, 4.15498943305707178695911210096, 4.35177636749552801250470686707, 5.55753287549069786827950034516, 6.04092095374430536601514260712, 6.57514151104056134418911092190, 7.61432299525255518283798725526, 8.483474537311554242253608351183
