| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 2.39·11-s + 12-s + 3.80·13-s + 16-s + 1.97·17-s − 18-s + 4.17·19-s + 2.39·22-s − 8.17·23-s − 24-s − 3.80·26-s + 27-s − 9.16·29-s − 1.60·31-s − 32-s − 2.39·33-s − 1.97·34-s + 36-s − 1.26·37-s − 4.17·38-s + 3.80·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.721·11-s + 0.288·12-s + 1.05·13-s + 0.250·16-s + 0.477·17-s − 0.235·18-s + 0.956·19-s + 0.510·22-s − 1.70·23-s − 0.204·24-s − 0.746·26-s + 0.192·27-s − 1.70·29-s − 0.288·31-s − 0.176·32-s − 0.416·33-s − 0.337·34-s + 0.166·36-s − 0.207·37-s − 0.676·38-s + 0.609·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2.39T + 11T^{2} \) |
| 13 | \( 1 - 3.80T + 13T^{2} \) |
| 17 | \( 1 - 1.97T + 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 + 8.17T + 23T^{2} \) |
| 29 | \( 1 + 9.16T + 29T^{2} \) |
| 31 | \( 1 + 1.60T + 31T^{2} \) |
| 37 | \( 1 + 1.26T + 37T^{2} \) |
| 41 | \( 1 - 3.19T + 41T^{2} \) |
| 43 | \( 1 + 4.90T + 43T^{2} \) |
| 47 | \( 1 - 3.00T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 5.77T + 67T^{2} \) |
| 71 | \( 1 - 6.94T + 71T^{2} \) |
| 73 | \( 1 + 0.506T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 1.89T + 83T^{2} \) |
| 89 | \( 1 + 1.97T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.75593855464765479421242163327, −7.18792629999822008587571337625, −6.08971026295000908171754856010, −5.73343029056914822413919903152, −4.67280752749937765559777576699, −3.63948034026087749679186085509, −3.16786674598820414997803035402, −2.07299413963275587738057283461, −1.38357817814304072286451750531, 0,
1.38357817814304072286451750531, 2.07299413963275587738057283461, 3.16786674598820414997803035402, 3.63948034026087749679186085509, 4.67280752749937765559777576699, 5.73343029056914822413919903152, 6.08971026295000908171754856010, 7.18792629999822008587571337625, 7.75593855464765479421242163327
