| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 5-s − 2·6-s + 4·8-s − 2·10-s − 11-s − 3·12-s − 13-s + 15-s + 5·16-s − 3·20-s − 2·22-s − 23-s − 4·24-s − 2·26-s − 29-s + 2·30-s − 31-s + 6·32-s + 33-s + 2·37-s + 39-s − 4·40-s − 41-s − 3·44-s + ⋯ |
| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 5-s − 2·6-s + 4·8-s − 2·10-s − 11-s − 3·12-s − 13-s + 15-s + 5·16-s − 3·20-s − 2·22-s − 23-s − 4·24-s − 2·26-s − 29-s + 2·30-s − 31-s + 6·32-s + 33-s + 2·37-s + 39-s − 4·40-s − 41-s − 3·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.152176303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.152176303\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 13 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 29 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 31 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 41 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 67 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 79 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 83 | $C_1$ | \( ( 1 - T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.08662681088524732658498686367, −11.92878601476356497669243678052, −11.48362480243374569351178297044, −11.15741192033290888839995195064, −10.64210747422095420115882841363, −10.29549945510468854334529785748, −9.726083695917808617414382926183, −8.845151764191126308860180953875, −7.85559919905754596078861043978, −7.64127581030885880174897304762, −7.41867145639848087708097846353, −6.65934243594218128434794578267, −5.86144625326211303585270062813, −5.85065420837689405939476020258, −5.12988544395750539868160378016, −4.68964727189577893116120082812, −4.07545421380961793180420069303, −3.56342714014302075545816589071, −2.72220946218321241717739633371, −2.04488031533664882038543243068,
2.04488031533664882038543243068, 2.72220946218321241717739633371, 3.56342714014302075545816589071, 4.07545421380961793180420069303, 4.68964727189577893116120082812, 5.12988544395750539868160378016, 5.85065420837689405939476020258, 5.86144625326211303585270062813, 6.65934243594218128434794578267, 7.41867145639848087708097846353, 7.64127581030885880174897304762, 7.85559919905754596078861043978, 8.845151764191126308860180953875, 9.726083695917808617414382926183, 10.29549945510468854334529785748, 10.64210747422095420115882841363, 11.15741192033290888839995195064, 11.48362480243374569351178297044, 11.92878601476356497669243678052, 12.08662681088524732658498686367
