L(s) = 1 | − 3-s − 5-s − 1.56·7-s + 9-s − 3.12·11-s + 2·13-s + 15-s + 0.438·17-s + 7.12·19-s + 1.56·21-s + 23-s + 25-s − 27-s + 4.43·29-s − 8.68·31-s + 3.12·33-s + 1.56·35-s − 3.56·37-s − 2·39-s + 7.56·41-s − 10.2·43-s − 45-s − 8·47-s − 4.56·49-s − 0.438·51-s − 3.56·53-s + 3.12·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.590·7-s + 0.333·9-s − 0.941·11-s + 0.554·13-s + 0.258·15-s + 0.106·17-s + 1.63·19-s + 0.340·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.824·29-s − 1.55·31-s + 0.543·33-s + 0.263·35-s − 0.585·37-s − 0.320·39-s + 1.18·41-s − 1.56·43-s − 0.149·45-s − 1.16·47-s − 0.651·49-s − 0.0613·51-s − 0.489·53-s + 0.421·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.006775668\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.006775668\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 0.438T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 + 8.68T + 31T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 41 | \( 1 - 7.56T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 3.56T + 53T^{2} \) |
| 59 | \( 1 + 2.43T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 1.56T + 67T^{2} \) |
| 71 | \( 1 - 0.684T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 6.24T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 5.12T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−7.940818418892383578697127069368, −7.50190400208261241211237728138, −6.70084149423345192745244626318, −6.00464279364606077396879937158, −5.22099792267254017349961293861, −4.69798395726745446966959343306, −3.45924325758709202291374021424, −3.13413402679821887791167082729, −1.73486531930053863556118194325, −0.55288318881786240260948979904,
0.55288318881786240260948979904, 1.73486531930053863556118194325, 3.13413402679821887791167082729, 3.45924325758709202291374021424, 4.69798395726745446966959343306, 5.22099792267254017349961293861, 6.00464279364606077396879937158, 6.70084149423345192745244626318, 7.50190400208261241211237728138, 7.940818418892383578697127069368