L(s) = 1 | − 2-s − 2.57·3-s + 4-s + 2.57·6-s + 3.57·7-s − 8-s + 3.61·9-s − 2.57·12-s + 5.30·13-s − 3.57·14-s + 16-s + 2.61·17-s − 3.61·18-s − 7.34·19-s − 9.18·21-s + 0.957·23-s + 2.57·24-s − 5.30·26-s − 1.58·27-s + 3.57·28-s + 10.3·29-s − 7.13·31-s − 32-s − 2.61·34-s + 3.61·36-s − 5.14·37-s + 7.34·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.48·3-s + 0.5·4-s + 1.04·6-s + 1.35·7-s − 0.353·8-s + 1.20·9-s − 0.742·12-s + 1.47·13-s − 0.954·14-s + 0.250·16-s + 0.634·17-s − 0.852·18-s − 1.68·19-s − 2.00·21-s + 0.199·23-s + 0.524·24-s − 1.04·26-s − 0.304·27-s + 0.675·28-s + 1.91·29-s − 1.28·31-s − 0.176·32-s − 0.448·34-s + 0.602·36-s − 0.845·37-s + 1.19·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2.57T + 3T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 13 | \( 1 - 5.30T + 13T^{2} \) |
| 17 | \( 1 - 2.61T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 - 0.957T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 7.13T + 31T^{2} \) |
| 37 | \( 1 + 5.14T + 37T^{2} \) |
| 41 | \( 1 + 4.56T + 41T^{2} \) |
| 43 | \( 1 + 0.356T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 1.56T + 53T^{2} \) |
| 59 | \( 1 + 3.66T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 5.56T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 + 7.73T + 73T^{2} \) |
| 79 | \( 1 + 2.32T + 79T^{2} \) |
| 83 | \( 1 + 1.14T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.76459023009820141596717277280, −6.91729039601572149778085404559, −6.24878067993634913231390191855, −5.77401478631350880618934372641, −4.88842443826166767909043478179, −4.36158357777290007738558574658, −3.19618822725633955805474448734, −1.73047943061840346613311235020, −1.25696738363587130270455720790, 0,
1.25696738363587130270455720790, 1.73047943061840346613311235020, 3.19618822725633955805474448734, 4.36158357777290007738558574658, 4.88842443826166767909043478179, 5.77401478631350880618934372641, 6.24878067993634913231390191855, 6.91729039601572149778085404559, 7.76459023009820141596717277280