| L(s) = 1 | − 5-s + 3·7-s − 8·17-s − 8·19-s + 25-s − 2·29-s − 6·31-s − 3·35-s − 5·41-s − 43-s − 5·47-s + 2·49-s − 8·53-s − 10·59-s + 7·61-s + 7·67-s − 14·71-s − 16·73-s − 10·79-s − 12·83-s + 8·85-s − 9·89-s + 8·95-s + 12·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s − 1.94·17-s − 1.83·19-s + 1/5·25-s − 0.371·29-s − 1.07·31-s − 0.507·35-s − 0.780·41-s − 0.152·43-s − 0.729·47-s + 2/7·49-s − 1.09·53-s − 1.30·59-s + 0.896·61-s + 0.855·67-s − 1.66·71-s − 1.87·73-s − 1.12·79-s − 1.31·83-s + 0.867·85-s − 0.953·89-s + 0.820·95-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2964123226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2964123226\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.98916461507690, −13.27858887136401, −12.87155746112380, −12.62909533170985, −11.66138994900278, −11.41748295128410, −11.05040229131150, −10.55901083474553, −10.06425273680570, −9.179837831635889, −8.778667927498742, −8.432875599277081, −7.940950335541198, −7.259157664891349, −6.827503032974617, −6.264748897028920, −5.655412953821970, −4.893339914281054, −4.404075520872665, −4.215415785806540, −3.332420302769349, −2.568878013957965, −1.831753732043101, −1.577510861414626, −0.1628418465714272,
0.1628418465714272, 1.577510861414626, 1.831753732043101, 2.568878013957965, 3.332420302769349, 4.215415785806540, 4.404075520872665, 4.893339914281054, 5.655412953821970, 6.264748897028920, 6.827503032974617, 7.259157664891349, 7.940950335541198, 8.432875599277081, 8.778667927498742, 9.179837831635889, 10.06425273680570, 10.55901083474553, 11.05040229131150, 11.41748295128410, 11.66138994900278, 12.62909533170985, 12.87155746112380, 13.27858887136401, 13.98916461507690
