| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s − 4·13-s − 4·14-s + 16-s + 17-s + 4·19-s − 6·23-s − 5·25-s − 4·26-s − 4·28-s + 10·29-s − 8·31-s + 32-s + 34-s − 10·37-s + 4·38-s + 2·41-s − 8·43-s − 6·46-s − 12·47-s + 9·49-s − 5·50-s − 4·52-s + 2·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s − 1.10·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s − 1.25·23-s − 25-s − 0.784·26-s − 0.755·28-s + 1.85·29-s − 1.43·31-s + 0.176·32-s + 0.171·34-s − 1.64·37-s + 0.648·38-s + 0.312·41-s − 1.21·43-s − 0.884·46-s − 1.75·47-s + 9/7·49-s − 0.707·50-s − 0.554·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.376245403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376245403\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.75932700392934, −14.30778651164578, −13.81630068675290, −13.30436181743314, −12.81805557454383, −12.28003303568983, −11.84302648023439, −11.55275878348283, −10.37195978687699, −10.18001123551061, −9.744636606054345, −9.152166722069494, −8.371725185799303, −7.701870410367432, −7.168081015946058, −6.581788925072571, −6.205624093083670, −5.353533376626631, −5.091545551075899, −4.133162903567141, −3.534379622794496, −3.116157910993544, −2.368525478536294, −1.638255645303948, −0.3627953656509338,
0.3627953656509338, 1.638255645303948, 2.368525478536294, 3.116157910993544, 3.534379622794496, 4.133162903567141, 5.091545551075899, 5.353533376626631, 6.205624093083670, 6.581788925072571, 7.168081015946058, 7.701870410367432, 8.371725185799303, 9.152166722069494, 9.744636606054345, 10.18001123551061, 10.37195978687699, 11.55275878348283, 11.84302648023439, 12.28003303568983, 12.81805557454383, 13.30436181743314, 13.81630068675290, 14.30778651164578, 14.75932700392934
