| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s − 5·11-s + 12-s − 6·13-s − 14-s + 16-s − 8·17-s + 18-s − 6·19-s − 21-s − 5·22-s + 24-s − 6·26-s + 27-s − 28-s − 29-s − 9·31-s + 32-s − 5·33-s − 8·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.50·11-s + 0.288·12-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 1.94·17-s + 0.235·18-s − 1.37·19-s − 0.218·21-s − 1.06·22-s + 0.204·24-s − 1.17·26-s + 0.192·27-s − 0.188·28-s − 0.185·29-s − 1.61·31-s + 0.176·32-s − 0.870·33-s − 1.37·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−14.76817815629314, −13.90296179512936, −13.40994520443112, −12.94070099300938, −12.77254742976231, −12.33418734432245, −11.43888979857645, −11.05640155935512, −10.50209413988587, −10.14579955523178, −9.312290316027414, −9.152879583473509, −8.260620315123587, −7.852503080512259, −7.359227497536772, −6.708469872824325, −6.425362326898388, −5.498627237612157, −5.097299874947291, −4.403716089415480, −4.179063040554130, −3.208963490842869, −2.601335372538621, −2.311745489715740, −1.732378101119729, 0, 0,
1.732378101119729, 2.311745489715740, 2.601335372538621, 3.208963490842869, 4.179063040554130, 4.403716089415480, 5.097299874947291, 5.498627237612157, 6.425362326898388, 6.708469872824325, 7.359227497536772, 7.852503080512259, 8.260620315123587, 9.152879583473509, 9.312290316027414, 10.14579955523178, 10.50209413988587, 11.05640155935512, 11.43888979857645, 12.33418734432245, 12.77254742976231, 12.94070099300938, 13.40994520443112, 13.90296179512936, 14.76817815629314
