L(s) = 1 | + 2-s + 2.44·3-s + 4-s + 2.44·6-s − 0.449·7-s + 8-s + 2.99·9-s + 2·11-s + 2.44·12-s + 2.44·13-s − 0.449·14-s + 16-s − 2·17-s + 2.99·18-s + 1.55·19-s − 1.10·21-s + 2·22-s + 2.44·23-s + 2.44·24-s + 2.44·26-s − 0.449·28-s − 29-s − 3·31-s + 32-s + 4.89·33-s − 2·34-s + 2.99·36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.41·3-s + 0.5·4-s + 0.999·6-s − 0.169·7-s + 0.353·8-s + 0.999·9-s + 0.603·11-s + 0.707·12-s + 0.679·13-s − 0.120·14-s + 0.250·16-s − 0.485·17-s + 0.707·18-s + 0.355·19-s − 0.240·21-s + 0.426·22-s + 0.510·23-s + 0.499·24-s + 0.480·26-s − 0.0849·28-s − 0.185·29-s − 0.538·31-s + 0.176·32-s + 0.852·33-s − 0.342·34-s + 0.499·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.334434871\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.334434871\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 7 | \( 1 + 0.449T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 1.55T + 19T^{2} \) |
| 23 | \( 1 - 2.44T + 23T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 1.44T + 37T^{2} \) |
| 41 | \( 1 + 3.34T + 41T^{2} \) |
| 43 | \( 1 - 0.898T + 43T^{2} \) |
| 47 | \( 1 + 3.89T + 47T^{2} \) |
| 53 | \( 1 - 5.55T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 1.44T + 61T^{2} \) |
| 67 | \( 1 + 8.55T + 67T^{2} \) |
| 71 | \( 1 + 3.34T + 71T^{2} \) |
| 73 | \( 1 + 1.34T + 73T^{2} \) |
| 79 | \( 1 - 6.89T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 8.44T + 89T^{2} \) |
| 97 | \( 1 + 7.34T + 97T^{2} \) |
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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
−9.311753608471028581414104698620, −8.756836302851105131567661498482, −7.954409836912874741843574240269, −7.10005303441772528088530094933, −6.33381230015963997667651606586, −5.25658347568770090225807512002, −4.11021732437097331400026782464, −3.49093896002718946503286362226, −2.62898818200158945230748081100, −1.53956994707840171168269220470,
1.53956994707840171168269220470, 2.62898818200158945230748081100, 3.49093896002718946503286362226, 4.11021732437097331400026782464, 5.25658347568770090225807512002, 6.33381230015963997667651606586, 7.10005303441772528088530094933, 7.954409836912874741843574240269, 8.756836302851105131567661498482, 9.311753608471028581414104698620