| L(s) = 1 | + 0.643·2-s − 1.59·3-s − 1.58·4-s − 1.02·6-s − 4.67·7-s − 2.30·8-s − 0.464·9-s + 5.33·11-s + 2.52·12-s + 1.14·13-s − 3.01·14-s + 1.68·16-s − 1.54·17-s − 0.299·18-s + 0.540·19-s + 7.44·21-s + 3.43·22-s + 0.709·23-s + 3.67·24-s + 0.740·26-s + 5.51·27-s + 7.41·28-s + 6.60·29-s + 3.00·31-s + 5.70·32-s − 8.49·33-s − 0.995·34-s + ⋯ |
| L(s) = 1 | + 0.455·2-s − 0.919·3-s − 0.792·4-s − 0.418·6-s − 1.76·7-s − 0.816·8-s − 0.154·9-s + 1.60·11-s + 0.728·12-s + 0.318·13-s − 0.805·14-s + 0.420·16-s − 0.374·17-s − 0.0705·18-s + 0.124·19-s + 1.62·21-s + 0.732·22-s + 0.148·23-s + 0.750·24-s + 0.145·26-s + 1.06·27-s + 1.40·28-s + 1.22·29-s + 0.539·31-s + 1.00·32-s − 1.47·33-s − 0.170·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 151 | \( 1 - T \) |
| good | 2 | \( 1 - 0.643T + 2T^{2} \) |
| 3 | \( 1 + 1.59T + 3T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 11 | \( 1 - 5.33T + 11T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 + 1.54T + 17T^{2} \) |
| 19 | \( 1 - 0.540T + 19T^{2} \) |
| 23 | \( 1 - 0.709T + 23T^{2} \) |
| 29 | \( 1 - 6.60T + 29T^{2} \) |
| 31 | \( 1 - 3.00T + 31T^{2} \) |
| 37 | \( 1 + 5.61T + 37T^{2} \) |
| 41 | \( 1 - 0.0821T + 41T^{2} \) |
| 43 | \( 1 - 0.525T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 7.61T + 61T^{2} \) |
| 67 | \( 1 - 7.44T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 8.01T + 73T^{2} \) |
| 79 | \( 1 + 0.296T + 79T^{2} \) |
| 83 | \( 1 - 4.92T + 83T^{2} \) |
| 89 | \( 1 + 7.48T + 89T^{2} \) |
| 97 | \( 1 + 2.88T + 97T^{2} \) |
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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
−8.399493564137022102741224493739, −6.90636596670695548356072651244, −6.43813375925604574264452811832, −6.04068962473439252521349717748, −5.16035733814532072794085238905, −4.29927968077728704150325500067, −3.56558014400290998184936873181, −2.89213968781971021736620824763, −1.03662225241511752658117406783, 0,
1.03662225241511752658117406783, 2.89213968781971021736620824763, 3.56558014400290998184936873181, 4.29927968077728704150325500067, 5.16035733814532072794085238905, 6.04068962473439252521349717748, 6.43813375925604574264452811832, 6.90636596670695548356072651244, 8.399493564137022102741224493739
