| L(s) = 1 | + 2.75·2-s − 3-s + 5.56·4-s + 4.15·5-s − 2.75·6-s + 7-s + 9.79·8-s + 9-s + 11.4·10-s − 1.99·11-s − 5.56·12-s − 0.629·13-s + 2.75·14-s − 4.15·15-s + 15.8·16-s + 2.75·18-s + 0.282·19-s + 23.1·20-s − 21-s − 5.49·22-s − 0.0258·23-s − 9.79·24-s + 12.2·25-s − 1.73·26-s − 27-s + 5.56·28-s − 6.72·29-s + ⋯ |
| L(s) = 1 | + 1.94·2-s − 0.577·3-s + 2.78·4-s + 1.85·5-s − 1.12·6-s + 0.377·7-s + 3.46·8-s + 0.333·9-s + 3.61·10-s − 0.602·11-s − 1.60·12-s − 0.174·13-s + 0.735·14-s − 1.07·15-s + 3.95·16-s + 0.648·18-s + 0.0647·19-s + 5.16·20-s − 0.218·21-s − 1.17·22-s − 0.00538·23-s − 2.00·24-s + 2.45·25-s − 0.339·26-s − 0.192·27-s + 1.05·28-s − 1.24·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.327609453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.327609453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.75T + 2T^{2} \) |
| 5 | \( 1 - 4.15T + 5T^{2} \) |
| 11 | \( 1 + 1.99T + 11T^{2} \) |
| 13 | \( 1 + 0.629T + 13T^{2} \) |
| 19 | \( 1 - 0.282T + 19T^{2} \) |
| 23 | \( 1 + 0.0258T + 23T^{2} \) |
| 29 | \( 1 + 6.72T + 29T^{2} \) |
| 31 | \( 1 - 1.98T + 31T^{2} \) |
| 37 | \( 1 + 9.69T + 37T^{2} \) |
| 41 | \( 1 + 7.39T + 41T^{2} \) |
| 43 | \( 1 + 0.0357T + 43T^{2} \) |
| 47 | \( 1 - 9.07T + 47T^{2} \) |
| 53 | \( 1 + 1.95T + 53T^{2} \) |
| 59 | \( 1 + 3.38T + 59T^{2} \) |
| 61 | \( 1 - 9.41T + 61T^{2} \) |
| 67 | \( 1 + 9.23T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 7.25T + 73T^{2} \) |
| 79 | \( 1 + 4.05T + 79T^{2} \) |
| 83 | \( 1 - 9.98T + 83T^{2} \) |
| 89 | \( 1 + 7.90T + 89T^{2} \) |
| 97 | \( 1 + 2.75T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.59380881436226125036775766048, −6.92286606860833975776885629752, −6.33566022782581522297441442660, −5.62658118096466944731166609638, −5.30494107789214006678183088276, −4.78975463379427311731753492526, −3.77767613203039724045421570811, −2.80388043897202427749238017617, −2.07967598502090200443528636740, −1.45747943443123399419939388227,
1.45747943443123399419939388227, 2.07967598502090200443528636740, 2.80388043897202427749238017617, 3.77767613203039724045421570811, 4.78975463379427311731753492526, 5.30494107789214006678183088276, 5.62658118096466944731166609638, 6.33566022782581522297441442660, 6.92286606860833975776885629752, 7.59380881436226125036775766048
