| L(s) = 1 | + 3-s − 1.23·5-s − 4·7-s + 9-s + 2.47·11-s + 4.47·13-s − 1.23·15-s − 5.23·17-s − 4·21-s − 3.23·23-s − 3.47·25-s + 27-s + 3.70·29-s − 10.4·31-s + 2.47·33-s + 4.94·35-s + 37-s + 4.47·39-s + 6.94·41-s − 6.47·43-s − 1.23·45-s − 8·47-s + 9·49-s − 5.23·51-s − 0.472·53-s − 3.05·55-s − 12.1·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.552·5-s − 1.51·7-s + 0.333·9-s + 0.745·11-s + 1.24·13-s − 0.319·15-s − 1.26·17-s − 0.872·21-s − 0.674·23-s − 0.694·25-s + 0.192·27-s + 0.688·29-s − 1.88·31-s + 0.430·33-s + 0.835·35-s + 0.164·37-s + 0.716·39-s + 1.08·41-s − 0.986·43-s − 0.184·45-s − 1.16·47-s + 1.28·49-s − 0.733·51-s − 0.0648·53-s − 0.412·55-s − 1.58·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 - 4.94T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 5.52T + 83T^{2} \) |
| 89 | \( 1 - 4.29T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−9.073491005058458439177178502234, −8.198820034706631637459994629925, −7.29873246903326547244149324965, −6.44646896111104113709908605551, −6.00664824422648962354888998537, −4.41354789052438555110249842496, −3.72342855511323619971568838734, −3.08819877776585365344193433766, −1.72854974527958647596139633553, 0,
1.72854974527958647596139633553, 3.08819877776585365344193433766, 3.72342855511323619971568838734, 4.41354789052438555110249842496, 6.00664824422648962354888998537, 6.44646896111104113709908605551, 7.29873246903326547244149324965, 8.198820034706631637459994629925, 9.073491005058458439177178502234
