| L(s) = 1 | + 0.473·2-s − 1.77·4-s − 3.75·5-s + 7-s − 1.78·8-s − 1.78·10-s − 2.70·13-s + 0.473·14-s + 2.70·16-s + 2.48·17-s + 1.74·19-s + 6.67·20-s − 3.79·23-s + 9.12·25-s − 1.28·26-s − 1.77·28-s − 5.80·29-s + 2.24·31-s + 4.85·32-s + 1.17·34-s − 3.75·35-s − 7.65·37-s + 0.826·38-s + 6.72·40-s − 4.18·41-s − 7.75·43-s − 1.79·46-s + ⋯ |
| L(s) = 1 | + 0.334·2-s − 0.887·4-s − 1.68·5-s + 0.377·7-s − 0.632·8-s − 0.563·10-s − 0.750·13-s + 0.126·14-s + 0.675·16-s + 0.602·17-s + 0.400·19-s + 1.49·20-s − 0.792·23-s + 1.82·25-s − 0.251·26-s − 0.335·28-s − 1.07·29-s + 0.403·31-s + 0.858·32-s + 0.201·34-s − 0.635·35-s − 1.25·37-s + 0.134·38-s + 1.06·40-s − 0.653·41-s − 1.18·43-s − 0.265·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5154689911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5154689911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.473T + 2T^{2} \) |
| 5 | \( 1 + 3.75T + 5T^{2} \) |
| 13 | \( 1 + 2.70T + 13T^{2} \) |
| 17 | \( 1 - 2.48T + 17T^{2} \) |
| 19 | \( 1 - 1.74T + 19T^{2} \) |
| 23 | \( 1 + 3.79T + 23T^{2} \) |
| 29 | \( 1 + 5.80T + 29T^{2} \) |
| 31 | \( 1 - 2.24T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 + 4.18T + 41T^{2} \) |
| 43 | \( 1 + 7.75T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 8.83T + 53T^{2} \) |
| 59 | \( 1 + 4.54T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 7.75T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 7.05T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 1.69T + 89T^{2} \) |
| 97 | \( 1 - 6.99T + 97T^{2} \) |
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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
−7.82515418302844249596193200578, −7.46252836269000117305399411014, −6.53465704927278254566553301206, −5.52489459987720916952163510312, −4.83738508243455063899341739184, −4.42903539436571908517550665317, −3.44922330681006592402191341817, −3.26084888123219521460214528746, −1.68407714579041391185556476754, −0.34266297581859492527457302155,
0.34266297581859492527457302155, 1.68407714579041391185556476754, 3.26084888123219521460214528746, 3.44922330681006592402191341817, 4.42903539436571908517550665317, 4.83738508243455063899341739184, 5.52489459987720916952163510312, 6.53465704927278254566553301206, 7.46252836269000117305399411014, 7.82515418302844249596193200578
