| L(s) = 1 | + 2.50·2-s + 2.28·3-s + 4.28·4-s − 1.22·5-s + 5.72·6-s + 0.778·7-s + 5.72·8-s + 2.22·9-s − 3.06·10-s + 11-s + 9.79·12-s + 1.95·14-s − 2.79·15-s + 5.79·16-s − 4.28·17-s + 5.57·18-s + 8.29·19-s − 5.23·20-s + 1.77·21-s + 2.50·22-s − 2.72·23-s + 13.0·24-s − 3.50·25-s − 1.77·27-s + 3.33·28-s + 7.36·29-s − 6.99·30-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 1.31·3-s + 2.14·4-s − 0.546·5-s + 2.33·6-s + 0.294·7-s + 2.02·8-s + 0.740·9-s − 0.968·10-s + 0.301·11-s + 2.82·12-s + 0.521·14-s − 0.720·15-s + 1.44·16-s − 1.03·17-s + 1.31·18-s + 1.90·19-s − 1.17·20-s + 0.388·21-s + 0.534·22-s − 0.569·23-s + 2.67·24-s − 0.701·25-s − 0.342·27-s + 0.630·28-s + 1.36·29-s − 1.27·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.314897164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.314897164\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.50T + 2T^{2} \) |
| 3 | \( 1 - 2.28T + 3T^{2} \) |
| 5 | \( 1 + 1.22T + 5T^{2} \) |
| 7 | \( 1 - 0.778T + 7T^{2} \) |
| 17 | \( 1 + 4.28T + 17T^{2} \) |
| 19 | \( 1 - 8.29T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 - 7.36T + 29T^{2} \) |
| 31 | \( 1 + 1.44T + 31T^{2} \) |
| 37 | \( 1 + 0.792T + 37T^{2} \) |
| 41 | \( 1 - 9.74T + 41T^{2} \) |
| 43 | \( 1 + 9.67T + 43T^{2} \) |
| 47 | \( 1 + 3.38T + 47T^{2} \) |
| 53 | \( 1 - 1.79T + 53T^{2} \) |
| 59 | \( 1 - 5.20T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 - 6.06T + 67T^{2} \) |
| 71 | \( 1 + 6.50T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 4.77T + 89T^{2} \) |
| 97 | \( 1 + 4.14T + 97T^{2} \) |
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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
−9.153256188616299262308871582066, −8.200157147672622949649408910913, −7.58818528939211274203573608161, −6.81597504391320519435015230011, −5.87782947598657706859839374689, −4.88040697011304291181724902051, −4.16019241928644333285525712340, −3.40418440105217952052430758849, −2.76766767160069132267668937654, −1.74135736969379652233449866278,
1.74135736969379652233449866278, 2.76766767160069132267668937654, 3.40418440105217952052430758849, 4.16019241928644333285525712340, 4.88040697011304291181724902051, 5.87782947598657706859839374689, 6.81597504391320519435015230011, 7.58818528939211274203573608161, 8.200157147672622949649408910913, 9.153256188616299262308871582066
