L(s) = 1 | − 2.44·3-s − 7-s + 2.99·9-s + 4.89·11-s − 4.44·13-s − 2·17-s − 1.55·19-s + 2.44·21-s + 2.89·23-s + 6.89·29-s − 8.89·31-s − 11.9·33-s − 2·37-s + 10.8·39-s − 1.10·41-s − 0.898·43-s + 8.89·47-s + 49-s + 4.89·51-s + 10.8·53-s + 3.79·57-s + 1.55·59-s + 3.55·61-s − 2.99·63-s − 8·67-s − 7.10·69-s + 1.10·71-s + ⋯ |
L(s) = 1 | − 1.41·3-s − 0.377·7-s + 0.999·9-s + 1.47·11-s − 1.23·13-s − 0.485·17-s − 0.355·19-s + 0.534·21-s + 0.604·23-s + 1.28·29-s − 1.59·31-s − 2.08·33-s − 0.328·37-s + 1.74·39-s − 0.171·41-s − 0.137·43-s + 1.29·47-s + 0.142·49-s + 0.685·51-s + 1.49·53-s + 0.503·57-s + 0.201·59-s + 0.454·61-s − 0.377·63-s − 0.977·67-s − 0.854·69-s + 0.130·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 1.55T + 19T^{2} \) |
| 23 | \( 1 - 2.89T + 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 + 8.89T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 + 0.898T + 43T^{2} \) |
| 47 | \( 1 - 8.89T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 1.10T + 71T^{2} \) |
| 73 | \( 1 + 2.89T + 73T^{2} \) |
| 79 | \( 1 + 6.89T + 79T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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
−8.615231778091495348346849328066, −7.17163881570688473865439000383, −6.92810538526476435072768523651, −6.12007128634609921791500494127, −5.39706195677561815522561806423, −4.60093446893466198134098845149, −3.85199279054058084326272293738, −2.54313912566700600959037110610, −1.21552053391431321987771072747, 0,
1.21552053391431321987771072747, 2.54313912566700600959037110610, 3.85199279054058084326272293738, 4.60093446893466198134098845149, 5.39706195677561815522561806423, 6.12007128634609921791500494127, 6.92810538526476435072768523651, 7.17163881570688473865439000383, 8.615231778091495348346849328066