| L(s) = 1 | − 0.741i·2-s + i·3-s + 1.45·4-s + 0.741·6-s − 1.03i·7-s − 2.55i·8-s − 9-s − 0.513·11-s + 1.45i·12-s + 3.54i·13-s − 0.767·14-s + 1.00·16-s + 1.36i·17-s + 0.741i·18-s − 0.894·19-s + ⋯ |
| L(s) = 1 | − 0.524i·2-s + 0.577i·3-s + 0.725·4-s + 0.302·6-s − 0.391i·7-s − 0.904i·8-s − 0.333·9-s − 0.154·11-s + 0.418i·12-s + 0.982i·13-s − 0.205·14-s + 0.251·16-s + 0.331i·17-s + 0.174i·18-s − 0.205·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.185297587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.185297587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.741iT - 2T^{2} \) |
| 7 | \( 1 + 1.03iT - 7T^{2} \) |
| 11 | \( 1 + 0.513T + 11T^{2} \) |
| 13 | \( 1 - 3.54iT - 13T^{2} \) |
| 17 | \( 1 - 1.36iT - 17T^{2} \) |
| 19 | \( 1 + 0.894T + 19T^{2} \) |
| 23 | \( 1 - 5.45iT - 23T^{2} \) |
| 29 | \( 1 - 9.65T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 2.19iT - 37T^{2} \) |
| 41 | \( 1 - 3.12T + 41T^{2} \) |
| 43 | \( 1 - 10.2iT - 43T^{2} \) |
| 47 | \( 1 + 7.65iT - 47T^{2} \) |
| 53 | \( 1 + 10.5iT - 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 + 8.80iT - 67T^{2} \) |
| 71 | \( 1 - 5.00T + 71T^{2} \) |
| 73 | \( 1 - 5.82iT - 73T^{2} \) |
| 79 | \( 1 - 6.74T + 79T^{2} \) |
| 83 | \( 1 + 7.99iT - 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 7.27iT - 97T^{2} \) |
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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
−9.535182261327862961670359857232, −8.485161907302991200497078402500, −7.71713502231419083366455017455, −6.67757876115146088937301784960, −6.24676106076238012137670468934, −4.96097346779163839730209683440, −4.15280885557689160090719274347, −3.27453383862741305081964278623, −2.34400982574309675230329214598, −1.14095921481711455022488035403,
0.945465214052967060389139513359, 2.48001786178365779583199505014, 2.86860308455560831644983852211, 4.50427879048544107062515416438, 5.47014617696274031939674818262, 6.19859148594978964780206344567, 6.77899116158205142450485132717, 7.66724955272173809994382576157, 8.251369527946548423114263825423, 8.879847171271022909247435267400
