| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 5·7-s + 8-s + 9-s + 10-s + 3·11-s − 12-s + 13-s − 5·14-s − 15-s + 16-s − 2·17-s + 18-s + 20-s + 5·21-s + 3·22-s + 9·23-s − 24-s + 25-s + 26-s − 27-s − 5·28-s − 4·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.88·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s − 0.288·12-s + 0.277·13-s − 1.33·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.223·20-s + 1.09·21-s + 0.639·22-s + 1.87·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.944·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.798279291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.798279291\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.25354988117923, −13.04308515119944, −12.57251958133836, −12.05541743130554, −11.61146445828836, −10.93597481777227, −10.75218510491214, −9.982870667749704, −9.547029570388801, −9.271229970947540, −8.717417113621792, −7.922533792569094, −7.137416179628094, −6.739090580027595, −6.506424028899384, −5.968979850160703, −5.604660154673221, −4.832917891202977, −4.331836338165861, −3.766147607980989, −3.140788687170075, −2.771827578829255, −2.025178243674666, −1.046718757287955, −0.6228040488521127,
0.6228040488521127, 1.046718757287955, 2.025178243674666, 2.771827578829255, 3.140788687170075, 3.766147607980989, 4.331836338165861, 4.832917891202977, 5.604660154673221, 5.968979850160703, 6.506424028899384, 6.739090580027595, 7.137416179628094, 7.922533792569094, 8.717417113621792, 9.271229970947540, 9.547029570388801, 9.982870667749704, 10.75218510491214, 10.93597481777227, 11.61146445828836, 12.05541743130554, 12.57251958133836, 13.04308515119944, 13.25354988117923
