L(s) = 1 | − 3-s − 5-s + 3.78·7-s + 9-s + 3.07·11-s − 5.07·13-s + 15-s + 3.28·17-s − 5.07·19-s − 3.78·21-s − 23-s + 25-s − 27-s + 1.28·29-s − 3.78·31-s − 3.07·33-s − 3.78·35-s + 3.22·37-s + 5.07·39-s − 9.42·41-s − 12.1·43-s − 45-s − 10.0·47-s + 7.35·49-s − 3.28·51-s + 5.84·53-s − 3.07·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.43·7-s + 0.333·9-s + 0.926·11-s − 1.40·13-s + 0.258·15-s + 0.796·17-s − 1.16·19-s − 0.826·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.238·29-s − 0.680·31-s − 0.534·33-s − 0.640·35-s + 0.529·37-s + 0.812·39-s − 1.47·41-s − 1.85·43-s − 0.149·45-s − 1.47·47-s + 1.05·49-s − 0.459·51-s + 0.803·53-s − 0.414·55-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 3.78T + 7T^{2} \) |
| 11 | \( 1 - 3.07T + 11T^{2} \) |
| 13 | \( 1 + 5.07T + 13T^{2} \) |
| 17 | \( 1 - 3.28T + 17T^{2} \) |
| 19 | \( 1 + 5.07T + 19T^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 + 3.78T + 31T^{2} \) |
| 37 | \( 1 - 3.22T + 37T^{2} \) |
| 41 | \( 1 + 9.42T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 5.84T + 53T^{2} \) |
| 59 | \( 1 - 2.29T + 59T^{2} \) |
| 61 | \( 1 + 3.07T + 61T^{2} \) |
| 67 | \( 1 + 3.22T + 67T^{2} \) |
| 71 | \( 1 + 7.28T + 71T^{2} \) |
| 73 | \( 1 - 9.63T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6.71T + 83T^{2} \) |
| 89 | \( 1 - 5.01T + 89T^{2} \) |
| 97 | \( 1 + 1.13T + 97T^{2} \) |
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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
−7.894393474584494341625434234438, −7.03347556699167426678120266592, −6.50881531120310829323965425972, −5.42898443117621763757481321457, −4.87711453049120629234004679352, −4.31605340502371419552091819719, −3.40727314929811459570682418444, −2.11005449893527353981997735215, −1.37710512594548165282861031508, 0,
1.37710512594548165282861031508, 2.11005449893527353981997735215, 3.40727314929811459570682418444, 4.31605340502371419552091819719, 4.87711453049120629234004679352, 5.42898443117621763757481321457, 6.50881531120310829323965425972, 7.03347556699167426678120266592, 7.894393474584494341625434234438