L(s) = 1 | − 3-s − 2.82·7-s + 9-s + 1.05·11-s + 13-s + 7.14·17-s − 6.61·19-s + 2.82·21-s − 3.49·23-s − 27-s + 4.82·29-s − 9.65·31-s − 1.05·33-s + 8.11·37-s − 39-s − 3.26·41-s − 7.44·43-s + 11.3·47-s + 1.00·49-s − 7.14·51-s + 4.53·53-s + 6.61·57-s + 1.05·59-s − 7.97·61-s − 2.82·63-s + 1.46·67-s + 3.49·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.06·7-s + 0.333·9-s + 0.318·11-s + 0.277·13-s + 1.73·17-s − 1.51·19-s + 0.617·21-s − 0.728·23-s − 0.192·27-s + 0.896·29-s − 1.73·31-s − 0.183·33-s + 1.33·37-s − 0.160·39-s − 0.510·41-s − 1.13·43-s + 1.65·47-s + 0.142·49-s − 1.00·51-s + 0.622·53-s + 0.876·57-s + 0.137·59-s − 1.02·61-s − 0.356·63-s + 0.179·67-s + 0.420·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.148822257\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.148822257\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 17 | \( 1 - 7.14T + 17T^{2} \) |
| 19 | \( 1 + 6.61T + 19T^{2} \) |
| 23 | \( 1 + 3.49T + 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + 9.65T + 31T^{2} \) |
| 37 | \( 1 - 8.11T + 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 + 7.44T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 - 1.05T + 59T^{2} \) |
| 61 | \( 1 + 7.97T + 61T^{2} \) |
| 67 | \( 1 - 1.46T + 67T^{2} \) |
| 71 | \( 1 - 0.845T + 71T^{2} \) |
| 73 | \( 1 - 9.28T + 73T^{2} \) |
| 79 | \( 1 - 6.85T + 79T^{2} \) |
| 83 | \( 1 - 7.97T + 83T^{2} \) |
| 89 | \( 1 - 0.139T + 89T^{2} \) |
| 97 | \( 1 + 6.93T + 97T^{2} \) |
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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
−7.85143408960233379667198895350, −6.98148600838988594213558531866, −6.41624974835728604358339133752, −5.86196652454025834886185857240, −5.23131975916466351113149239450, −4.14403714492357139085008656529, −3.66681168140864451469881512575, −2.75316715297478678991346418440, −1.66647291057730089026257462286, −0.55274360259801295630814271448,
0.55274360259801295630814271448, 1.66647291057730089026257462286, 2.75316715297478678991346418440, 3.66681168140864451469881512575, 4.14403714492357139085008656529, 5.23131975916466351113149239450, 5.86196652454025834886185857240, 6.41624974835728604358339133752, 6.98148600838988594213558531866, 7.85143408960233379667198895350