| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 4.16·7-s − 8-s + 9-s − 4.27·11-s + 12-s + 6.14·13-s + 4.16·14-s + 16-s − 5.15·17-s − 18-s + 7.61·19-s − 4.16·21-s + 4.27·22-s + 2.99·23-s − 24-s − 6.14·26-s + 27-s − 4.16·28-s − 5.67·29-s + 3.45·31-s − 32-s − 4.27·33-s + 5.15·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.57·7-s − 0.353·8-s + 0.333·9-s − 1.28·11-s + 0.288·12-s + 1.70·13-s + 1.11·14-s + 0.250·16-s − 1.25·17-s − 0.235·18-s + 1.74·19-s − 0.908·21-s + 0.910·22-s + 0.623·23-s − 0.204·24-s − 1.20·26-s + 0.192·27-s − 0.787·28-s − 1.05·29-s + 0.620·31-s − 0.176·32-s − 0.743·33-s + 0.884·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.16T + 7T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 - 6.14T + 13T^{2} \) |
| 17 | \( 1 + 5.15T + 17T^{2} \) |
| 19 | \( 1 - 7.61T + 19T^{2} \) |
| 23 | \( 1 - 2.99T + 23T^{2} \) |
| 29 | \( 1 + 5.67T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 + 7.53T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 3.38T + 43T^{2} \) |
| 47 | \( 1 - 0.632T + 47T^{2} \) |
| 53 | \( 1 + 6.34T + 53T^{2} \) |
| 59 | \( 1 + 3.03T + 59T^{2} \) |
| 61 | \( 1 + 2.29T + 61T^{2} \) |
| 67 | \( 1 + 0.726T + 67T^{2} \) |
| 71 | \( 1 - 3.20T + 71T^{2} \) |
| 73 | \( 1 + 9.06T + 73T^{2} \) |
| 79 | \( 1 + 6.62T + 79T^{2} \) |
| 83 | \( 1 + 3.20T + 83T^{2} \) |
| 89 | \( 1 + 4.59T + 89T^{2} \) |
| 97 | \( 1 + 9.16T + 97T^{2} \) |
| 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
−8.244324049671434519196746979335, −7.47185942179522490520767387354, −6.82499043928895885799975758099, −6.08760810541827916461393470862, −5.33238745755257280809383748619, −3.99130612133991251850097840226, −3.16403325876517513982872954565, −2.68001203199375485629480820539, −1.33218282045158082819987072289, 0,
1.33218282045158082819987072289, 2.68001203199375485629480820539, 3.16403325876517513982872954565, 3.99130612133991251850097840226, 5.33238745755257280809383748619, 6.08760810541827916461393470862, 6.82499043928895885799975758099, 7.47185942179522490520767387354, 8.244324049671434519196746979335
