| L(s) = 1 | + 2-s + 1.18·3-s + 4-s − 3.79·5-s + 1.18·6-s − 0.397·7-s + 8-s − 1.58·9-s − 3.79·10-s − 11-s + 1.18·12-s − 5.64·13-s − 0.397·14-s − 4.50·15-s + 16-s − 5.00·17-s − 1.58·18-s − 2.37·19-s − 3.79·20-s − 0.472·21-s − 22-s − 2.14·23-s + 1.18·24-s + 9.37·25-s − 5.64·26-s − 5.45·27-s − 0.397·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.686·3-s + 0.5·4-s − 1.69·5-s + 0.485·6-s − 0.150·7-s + 0.353·8-s − 0.528·9-s − 1.19·10-s − 0.301·11-s + 0.343·12-s − 1.56·13-s − 0.106·14-s − 1.16·15-s + 0.250·16-s − 1.21·17-s − 0.373·18-s − 0.543·19-s − 0.847·20-s − 0.103·21-s − 0.213·22-s − 0.447·23-s + 0.242·24-s + 1.87·25-s − 1.10·26-s − 1.04·27-s − 0.0751·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 - 1.18T + 3T^{2} \) |
| 5 | \( 1 + 3.79T + 5T^{2} \) |
| 7 | \( 1 + 0.397T + 7T^{2} \) |
| 13 | \( 1 + 5.64T + 13T^{2} \) |
| 17 | \( 1 + 5.00T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 23 | \( 1 + 2.14T + 23T^{2} \) |
| 29 | \( 1 - 9.88T + 29T^{2} \) |
| 31 | \( 1 - 5.01T + 31T^{2} \) |
| 37 | \( 1 - 9.71T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 47 | \( 1 + 4.77T + 47T^{2} \) |
| 53 | \( 1 - 0.138T + 53T^{2} \) |
| 59 | \( 1 + 8.19T + 59T^{2} \) |
| 61 | \( 1 + 3.44T + 61T^{2} \) |
| 67 | \( 1 + 1.05T + 67T^{2} \) |
| 71 | \( 1 + 2.88T + 71T^{2} \) |
| 73 | \( 1 - 0.937T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 5.28T + 83T^{2} \) |
| 89 | \( 1 + 4.31T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−9.604370376793048709515747644972, −8.425431949849190280549239608765, −8.027118512114529188748295578878, −7.16376014007798016050110008378, −6.29479205334476134176166870073, −4.74582606790531316416406586831, −4.36370673458752099346711211506, −3.13145003222705114234886346965, −2.50081596983752154423312482955, 0,
2.50081596983752154423312482955, 3.13145003222705114234886346965, 4.36370673458752099346711211506, 4.74582606790531316416406586831, 6.29479205334476134176166870073, 7.16376014007798016050110008378, 8.027118512114529188748295578878, 8.425431949849190280549239608765, 9.604370376793048709515747644972
