| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 4·7-s + 8-s + 9-s + 10-s − 12-s − 5·13-s − 4·14-s − 15-s + 16-s + 18-s + 20-s + 4·21-s − 24-s − 4·25-s − 5·26-s − 27-s − 4·28-s − 7·29-s − 30-s + 8·31-s + 32-s − 4·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 1.38·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.223·20-s + 0.872·21-s − 0.204·24-s − 4/5·25-s − 0.980·26-s − 0.192·27-s − 0.755·28-s − 1.29·29-s − 0.182·30-s + 1.43·31-s + 0.176·32-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.11935190730372, −12.86340714096355, −12.26127338063325, −12.11438279605182, −11.39586612457028, −11.13398587959821, −10.22873227469177, −10.04427076193482, −9.581838496375058, −9.412998509240063, −8.395682816813539, −7.981671589681213, −7.223571934035578, −6.881799231930083, −6.472771234110128, −6.045419304363968, −5.388355717696514, −5.185545002169226, −4.447923468726649, −3.922897845793858, −3.337751929211258, −2.818583851665021, −2.233631330659056, −1.679448081809324, −0.6496536280592860, 0,
0.6496536280592860, 1.679448081809324, 2.233631330659056, 2.818583851665021, 3.337751929211258, 3.922897845793858, 4.447923468726649, 5.185545002169226, 5.388355717696514, 6.045419304363968, 6.472771234110128, 6.881799231930083, 7.223571934035578, 7.981671589681213, 8.395682816813539, 9.412998509240063, 9.581838496375058, 10.04427076193482, 10.22873227469177, 11.13398587959821, 11.39586612457028, 12.11438279605182, 12.26127338063325, 12.86340714096355, 13.11935190730372
