| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 0.585·11-s − 12-s − 3·13-s + 16-s − 0.171·17-s + 18-s − 1.41·19-s + 0.585·22-s − 2.41·23-s − 24-s − 3·26-s − 27-s + 29-s + 0.414·31-s + 32-s − 0.585·33-s − 0.171·34-s + 36-s + 7.07·37-s − 1.41·38-s + 3·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.176·11-s − 0.288·12-s − 0.832·13-s + 0.250·16-s − 0.0416·17-s + 0.235·18-s − 0.324·19-s + 0.124·22-s − 0.503·23-s − 0.204·24-s − 0.588·26-s − 0.192·27-s + 0.185·29-s + 0.0743·31-s + 0.176·32-s − 0.101·33-s − 0.0294·34-s + 0.166·36-s + 1.16·37-s − 0.229·38-s + 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 0.585T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + 0.171T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 + 2.41T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 0.414T + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 + 0.171T + 41T^{2} \) |
| 43 | \( 1 + 4.41T + 43T^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 + 1.34T + 53T^{2} \) |
| 59 | \( 1 + 1.24T + 59T^{2} \) |
| 61 | \( 1 + 5.82T + 61T^{2} \) |
| 67 | \( 1 + 7.31T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 0.928T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−7.44405355316736223063804502174, −6.63751982469528925412081091255, −6.17748654935949412502055366768, −5.38434481456279028914926181174, −4.71719259190885492050295794517, −4.15813146499955283073291242915, −3.20551150068266142717276896923, −2.34858306802232441305385644393, −1.38815644151582075873748481269, 0,
1.38815644151582075873748481269, 2.34858306802232441305385644393, 3.20551150068266142717276896923, 4.15813146499955283073291242915, 4.71719259190885492050295794517, 5.38434481456279028914926181174, 6.17748654935949412502055366768, 6.63751982469528925412081091255, 7.44405355316736223063804502174
