L(s) = 1 | − 1.88·2-s + 1.56·4-s − 5-s + 7-s + 0.828·8-s + 1.88·10-s + 2.76·11-s + 5.73·13-s − 1.88·14-s − 4.68·16-s − 4.82·17-s + 4.18·19-s − 1.56·20-s − 5.21·22-s + 23-s + 25-s − 10.8·26-s + 1.56·28-s + 8.32·29-s + 5.10·31-s + 7.18·32-s + 9.10·34-s − 35-s + 3.98·37-s − 7.89·38-s − 0.828·40-s + 3.49·41-s + ⋯ |
L(s) = 1 | − 1.33·2-s + 0.780·4-s − 0.447·5-s + 0.377·7-s + 0.292·8-s + 0.596·10-s + 0.833·11-s + 1.58·13-s − 0.504·14-s − 1.17·16-s − 1.17·17-s + 0.959·19-s − 0.349·20-s − 1.11·22-s + 0.208·23-s + 0.200·25-s − 2.12·26-s + 0.294·28-s + 1.54·29-s + 0.916·31-s + 1.27·32-s + 1.56·34-s − 0.169·35-s + 0.655·37-s − 1.28·38-s − 0.131·40-s + 0.545·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.197331984\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.197331984\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 1.88T + 2T^{2} \) |
| 11 | \( 1 - 2.76T + 11T^{2} \) |
| 13 | \( 1 - 5.73T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 4.18T + 19T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 - 3.98T + 37T^{2} \) |
| 41 | \( 1 - 3.49T + 41T^{2} \) |
| 43 | \( 1 - 6.13T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 7.60T + 53T^{2} \) |
| 59 | \( 1 - 0.943T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 + 3.94T + 67T^{2} \) |
| 71 | \( 1 + 0.189T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 8.77T + 79T^{2} \) |
| 83 | \( 1 - 9.38T + 83T^{2} \) |
| 89 | \( 1 - 8.69T + 89T^{2} \) |
| 97 | \( 1 - 8.78T + 97T^{2} \) |
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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
−8.015156898169406976696051116798, −7.54693117336410235848295382271, −6.47261969702007751852003103713, −6.36552252642378244039126932679, −4.94869668599941702445492348100, −4.33901102655749146202791084260, −3.53948483253386712473034871341, −2.45195118606173719299496966482, −1.32385003995335249974383496825, −0.801726164730220970000361149343,
0.801726164730220970000361149343, 1.32385003995335249974383496825, 2.45195118606173719299496966482, 3.53948483253386712473034871341, 4.33901102655749146202791084260, 4.94869668599941702445492348100, 6.36552252642378244039126932679, 6.47261969702007751852003103713, 7.54693117336410235848295382271, 8.015156898169406976696051116798