| L(s) = 1 | − 4·5-s − 2·7-s + 2·13-s + 4·17-s + 6·19-s + 23-s + 11·25-s − 10·29-s − 4·31-s + 8·35-s − 2·37-s + 6·41-s + 6·43-s + 8·47-s − 3·49-s − 8·53-s − 4·59-s − 2·61-s − 8·65-s − 6·67-s − 14·73-s + 10·79-s + 16·83-s − 16·85-s − 4·91-s − 24·95-s − 18·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.755·7-s + 0.554·13-s + 0.970·17-s + 1.37·19-s + 0.208·23-s + 11/5·25-s − 1.85·29-s − 0.718·31-s + 1.35·35-s − 0.328·37-s + 0.937·41-s + 0.914·43-s + 1.16·47-s − 3/7·49-s − 1.09·53-s − 0.520·59-s − 0.256·61-s − 0.992·65-s − 0.733·67-s − 1.63·73-s + 1.12·79-s + 1.75·83-s − 1.73·85-s − 0.419·91-s − 2.46·95-s − 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
| 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
−7.982779653729005423796045463243, −7.60025327335934219686584914491, −7.04210848814266796650626171424, −5.96460038739869597324085593025, −5.21317893220903717464082428403, −4.09052236142486331024520258887, −3.56288163046526004578369727610, −2.94172601163278029965437630516, −1.19118767279990453620847477651, 0,
1.19118767279990453620847477651, 2.94172601163278029965437630516, 3.56288163046526004578369727610, 4.09052236142486331024520258887, 5.21317893220903717464082428403, 5.96460038739869597324085593025, 7.04210848814266796650626171424, 7.60025327335934219686584914491, 7.982779653729005423796045463243
