| L(s) = 1 | + 2-s + 3.07·3-s + 4-s + 3.07·6-s + 7-s + 8-s + 6.48·9-s − 3.48·11-s + 3.07·12-s + 14-s + 16-s + 17-s + 6.48·18-s + 7.48·19-s + 3.07·21-s − 3.48·22-s + 3.07·23-s + 3.07·24-s + 10.7·27-s + 28-s − 3.07·29-s − 7.63·31-s + 32-s − 10.7·33-s + 34-s + 6.48·36-s + 2.15·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.77·3-s + 0.5·4-s + 1.25·6-s + 0.377·7-s + 0.353·8-s + 2.16·9-s − 1.04·11-s + 0.888·12-s + 0.267·14-s + 0.250·16-s + 0.242·17-s + 1.52·18-s + 1.71·19-s + 0.671·21-s − 0.742·22-s + 0.642·23-s + 0.628·24-s + 2.06·27-s + 0.188·28-s − 0.571·29-s − 1.37·31-s + 0.176·32-s − 1.86·33-s + 0.171·34-s + 1.08·36-s + 0.354·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.807339198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.807339198\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 3.07T + 3T^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 19 | \( 1 - 7.48T + 19T^{2} \) |
| 23 | \( 1 - 3.07T + 23T^{2} \) |
| 29 | \( 1 + 3.07T + 29T^{2} \) |
| 31 | \( 1 + 7.63T + 31T^{2} \) |
| 37 | \( 1 - 2.15T + 37T^{2} \) |
| 41 | \( 1 - 0.519T + 41T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 - 3.07T + 47T^{2} \) |
| 53 | \( 1 + 5.07T + 53T^{2} \) |
| 59 | \( 1 + 1.07T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 0.803T + 79T^{2} \) |
| 83 | \( 1 + 6.31T + 83T^{2} \) |
| 89 | \( 1 + 2.80T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.912891768432854587614953887125, −7.48496356612550445630456650154, −7.04503218123805675438318552363, −5.69531720391768743398741391575, −5.15400616483355394749443022698, −4.24839447915048108012861667189, −3.48583819780453146350973386585, −2.89861621513994923530394471190, −2.22156737596747933076881967923, −1.25550269383113917042103029010,
1.25550269383113917042103029010, 2.22156737596747933076881967923, 2.89861621513994923530394471190, 3.48583819780453146350973386585, 4.24839447915048108012861667189, 5.15400616483355394749443022698, 5.69531720391768743398741391575, 7.04503218123805675438318552363, 7.48496356612550445630456650154, 7.912891768432854587614953887125
