L(s) = 1 | − 2.61·2-s + 4.85·4-s − 5-s − 7-s − 7.47·8-s + 2.61·10-s − 3.23·13-s + 2.61·14-s + 9.85·16-s − 8.09·17-s + 6.23·19-s − 4.85·20-s + 6.09·23-s − 4·25-s + 8.47·26-s − 4.85·28-s − 2.38·29-s + 0.236·31-s − 10.8·32-s + 21.1·34-s + 35-s − 2.47·37-s − 16.3·38-s + 7.47·40-s + 11.1·41-s − 7.56·43-s − 15.9·46-s + ⋯ |
L(s) = 1 | − 1.85·2-s + 2.42·4-s − 0.447·5-s − 0.377·7-s − 2.64·8-s + 0.827·10-s − 0.897·13-s + 0.699·14-s + 2.46·16-s − 1.96·17-s + 1.43·19-s − 1.08·20-s + 1.26·23-s − 0.800·25-s + 1.66·26-s − 0.917·28-s − 0.442·29-s + 0.0423·31-s − 1.91·32-s + 3.63·34-s + 0.169·35-s − 0.406·37-s − 2.64·38-s + 1.18·40-s + 1.74·41-s − 1.15·43-s − 2.35·46-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 8.09T + 17T^{2} \) |
| 19 | \( 1 - 6.23T + 19T^{2} \) |
| 23 | \( 1 - 6.09T + 23T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 - 0.236T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 7.56T + 43T^{2} \) |
| 47 | \( 1 - 4.38T + 47T^{2} \) |
| 53 | \( 1 - 4.61T + 53T^{2} \) |
| 59 | \( 1 - 0.0901T + 59T^{2} \) |
| 61 | \( 1 - 5.38T + 61T^{2} \) |
| 67 | \( 1 - 7.32T + 67T^{2} \) |
| 71 | \( 1 - 4.90T + 71T^{2} \) |
| 73 | \( 1 - 9.76T + 73T^{2} \) |
| 79 | \( 1 + 8.61T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 0.145T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−7.48085905236345556870915698685, −7.17704354450408090541549292219, −6.60010221302847682970177704701, −5.68268472682474777385594792950, −4.75039439926802169622314186159, −3.66468988033671951088633000857, −2.70026555537369215000132438275, −2.10319146656435429834705392838, −0.911068768582631026107892254663, 0,
0.911068768582631026107892254663, 2.10319146656435429834705392838, 2.70026555537369215000132438275, 3.66468988033671951088633000857, 4.75039439926802169622314186159, 5.68268472682474777385594792950, 6.60010221302847682970177704701, 7.17704354450408090541549292219, 7.48085905236345556870915698685