| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 11-s − 1.54·13-s − 15-s − 5.68·17-s + 1.43·19-s − 21-s + 4.93·23-s + 25-s + 27-s − 6.48·29-s + 7.04·31-s + 33-s + 35-s − 11.0·37-s − 1.54·39-s − 5.04·41-s + 6.51·43-s − 45-s + 8.66·47-s + 49-s − 5.68·51-s + 13.1·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 0.333·9-s + 0.301·11-s − 0.428·13-s − 0.258·15-s − 1.37·17-s + 0.330·19-s − 0.218·21-s + 1.02·23-s + 0.200·25-s + 0.192·27-s − 1.20·29-s + 1.26·31-s + 0.174·33-s + 0.169·35-s − 1.82·37-s − 0.247·39-s − 0.787·41-s + 0.993·43-s − 0.149·45-s + 1.26·47-s + 0.142·49-s − 0.796·51-s + 1.81·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.906927913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.906927913\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 1.54T + 13T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 - 1.43T + 19T^{2} \) |
| 23 | \( 1 - 4.93T + 23T^{2} \) |
| 29 | \( 1 + 6.48T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 5.04T + 41T^{2} \) |
| 43 | \( 1 - 6.51T + 43T^{2} \) |
| 47 | \( 1 - 8.66T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 5.60T + 59T^{2} \) |
| 61 | \( 1 + 3.43T + 61T^{2} \) |
| 67 | \( 1 + 5.87T + 67T^{2} \) |
| 71 | \( 1 + 8.16T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 5.92T + 79T^{2} \) |
| 83 | \( 1 - 5.30T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 9.98T + 97T^{2} \) |
| 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
−7.51516368929173136724977512319, −7.18323208933403723393232113637, −6.55215676858477493551535195537, −5.64749475547648723574507170848, −4.80602140446647100501375174828, −4.15126634179694960628861251672, −3.43313344965468164209222338777, −2.67851968552826380732448419159, −1.87109281781126388294620362036, −0.63098377039291375436848101442,
0.63098377039291375436848101442, 1.87109281781126388294620362036, 2.67851968552826380732448419159, 3.43313344965468164209222338777, 4.15126634179694960628861251672, 4.80602140446647100501375174828, 5.64749475547648723574507170848, 6.55215676858477493551535195537, 7.18323208933403723393232113637, 7.51516368929173136724977512319
