| L(s) = 1 | − 4·5-s − 4·7-s + 2·11-s − 13·13-s + 6·17-s + 36·19-s − 20·23-s − 109·25-s + 14·29-s + 152·31-s + 16·35-s − 258·37-s − 84·41-s + 188·43-s + 254·47-s − 327·49-s − 366·53-s − 8·55-s + 550·59-s − 14·61-s + 52·65-s − 448·67-s + 926·71-s + 254·73-s − 8·77-s − 1.32e3·79-s + 186·83-s + ⋯ |
| L(s) = 1 | − 0.357·5-s − 0.215·7-s + 0.0548·11-s − 0.277·13-s + 0.0856·17-s + 0.434·19-s − 0.181·23-s − 0.871·25-s + 0.0896·29-s + 0.880·31-s + 0.0772·35-s − 1.14·37-s − 0.319·41-s + 0.666·43-s + 0.788·47-s − 0.953·49-s − 0.948·53-s − 0.0196·55-s + 1.21·59-s − 0.0293·61-s + 0.0992·65-s − 0.816·67-s + 1.54·71-s + 0.407·73-s − 0.0118·77-s − 1.89·79-s + 0.245·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.562324554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.562324554\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 T + p^{3} T^{2} \) |
| 19 | \( 1 - 36 T + p^{3} T^{2} \) |
| 23 | \( 1 + 20 T + p^{3} T^{2} \) |
| 29 | \( 1 - 14 T + p^{3} T^{2} \) |
| 31 | \( 1 - 152 T + p^{3} T^{2} \) |
| 37 | \( 1 + 258 T + p^{3} T^{2} \) |
| 41 | \( 1 + 84 T + p^{3} T^{2} \) |
| 43 | \( 1 - 188 T + p^{3} T^{2} \) |
| 47 | \( 1 - 254 T + p^{3} T^{2} \) |
| 53 | \( 1 + 366 T + p^{3} T^{2} \) |
| 59 | \( 1 - 550 T + p^{3} T^{2} \) |
| 61 | \( 1 + 14 T + p^{3} T^{2} \) |
| 67 | \( 1 + 448 T + p^{3} T^{2} \) |
| 71 | \( 1 - 926 T + p^{3} T^{2} \) |
| 73 | \( 1 - 254 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1328 T + p^{3} T^{2} \) |
| 83 | \( 1 - 186 T + p^{3} T^{2} \) |
| 89 | \( 1 - 336 T + p^{3} T^{2} \) |
| 97 | \( 1 - 614 T + p^{3} T^{2} \) |
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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
−8.841614465449774591097367794189, −8.050558688452731367317300596239, −7.35595793386759923570433561316, −6.53270122199644278179487352995, −5.66856620484917032854878358407, −4.77233380532924688600863991681, −3.85850598492027860212521412144, −2.99744905823897510690205128855, −1.86453963850477286953316853525, −0.57581623357293209519882570977,
0.57581623357293209519882570977, 1.86453963850477286953316853525, 2.99744905823897510690205128855, 3.85850598492027860212521412144, 4.77233380532924688600863991681, 5.66856620484917032854878358407, 6.53270122199644278179487352995, 7.35595793386759923570433561316, 8.050558688452731367317300596239, 8.841614465449774591097367794189
