L(s) = 1 | + 5-s − 2·11-s − 13-s + 2·17-s − 2·19-s − 2·23-s + 25-s + 6·29-s − 2·31-s − 6·37-s + 2·41-s + 6·43-s − 8·47-s + 2·53-s − 2·55-s + 6·59-s + 14·61-s − 65-s − 10·71-s + 2·73-s − 4·79-s + 12·83-s + 2·85-s − 6·89-s − 2·95-s − 2·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.603·11-s − 0.277·13-s + 0.485·17-s − 0.458·19-s − 0.417·23-s + 1/5·25-s + 1.11·29-s − 0.359·31-s − 0.986·37-s + 0.312·41-s + 0.914·43-s − 1.16·47-s + 0.274·53-s − 0.269·55-s + 0.781·59-s + 1.79·61-s − 0.124·65-s − 1.18·71-s + 0.234·73-s − 0.450·79-s + 1.31·83-s + 0.216·85-s − 0.635·89-s − 0.205·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.964579050\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.964579050\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−12.74818989749117, −12.67727168946600, −11.98105083634540, −11.63311379228753, −10.99511276380931, −10.52946868469478, −10.10801770070140, −9.843002746312995, −9.131489828582319, −8.776256624179905, −8.075379570718556, −7.932833421164478, −7.137679462193752, −6.757571456725487, −6.266400978859418, −5.565350406278954, −5.335565390904329, −4.734746320812418, −4.122716663345505, −3.608097497812042, −2.863489564581391, −2.477949596284891, −1.847624528537283, −1.177986523206129, −0.3968172266590303,
0.3968172266590303, 1.177986523206129, 1.847624528537283, 2.477949596284891, 2.863489564581391, 3.608097497812042, 4.122716663345505, 4.734746320812418, 5.335565390904329, 5.565350406278954, 6.266400978859418, 6.757571456725487, 7.137679462193752, 7.932833421164478, 8.075379570718556, 8.776256624179905, 9.131489828582319, 9.843002746312995, 10.10801770070140, 10.52946868469478, 10.99511276380931, 11.63311379228753, 11.98105083634540, 12.67727168946600, 12.74818989749117