L(s) = 1 | + 2-s − 0.310·3-s + 4-s − 5-s − 0.310·6-s + 1.47·7-s + 8-s − 2.90·9-s − 10-s + 2.97·11-s − 0.310·12-s − 6.20·13-s + 1.47·14-s + 0.310·15-s + 16-s − 4.41·17-s − 2.90·18-s + 4.87·19-s − 20-s − 0.459·21-s + 2.97·22-s + 2.42·23-s − 0.310·24-s + 25-s − 6.20·26-s + 1.83·27-s + 1.47·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.179·3-s + 0.5·4-s − 0.447·5-s − 0.126·6-s + 0.559·7-s + 0.353·8-s − 0.967·9-s − 0.316·10-s + 0.896·11-s − 0.0897·12-s − 1.72·13-s + 0.395·14-s + 0.0802·15-s + 0.250·16-s − 1.07·17-s − 0.684·18-s + 1.11·19-s − 0.223·20-s − 0.100·21-s + 0.634·22-s + 0.504·23-s − 0.0634·24-s + 0.200·25-s − 1.21·26-s + 0.353·27-s + 0.279·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 - T \) |
good | 3 | \( 1 + 0.310T + 3T^{2} \) |
| 7 | \( 1 - 1.47T + 7T^{2} \) |
| 11 | \( 1 - 2.97T + 11T^{2} \) |
| 13 | \( 1 + 6.20T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 - 4.87T + 19T^{2} \) |
| 23 | \( 1 - 2.42T + 23T^{2} \) |
| 29 | \( 1 - 2.62T + 29T^{2} \) |
| 31 | \( 1 + 3.97T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 9.83T + 53T^{2} \) |
| 59 | \( 1 + 6.88T + 59T^{2} \) |
| 61 | \( 1 - 4.95T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 2.88T + 71T^{2} \) |
| 73 | \( 1 + 7.23T + 73T^{2} \) |
| 79 | \( 1 - 9.98T + 79T^{2} \) |
| 83 | \( 1 - 0.667T + 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 + 5.27T + 97T^{2} \) |
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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
−7.65897156499969744245859378776, −6.92750909751821447903503166311, −6.31175726067322915028664941720, −5.38045484322839951928535880678, −4.81513009874036536857795778553, −4.25987462890362308784205152937, −3.16416876768140210225440153566, −2.58177229269746697563730742907, −1.45004477360020146825886176944, 0,
1.45004477360020146825886176944, 2.58177229269746697563730742907, 3.16416876768140210225440153566, 4.25987462890362308784205152937, 4.81513009874036536857795778553, 5.38045484322839951928535880678, 6.31175726067322915028664941720, 6.92750909751821447903503166311, 7.65897156499969744245859378776