| L(s) = 1 | + 1.05·2-s − 0.877·4-s + 1.30·5-s + 0.720·7-s − 3.04·8-s + 1.38·10-s − 4.26·11-s + 6.49·13-s + 0.763·14-s − 1.47·16-s − 1.60·17-s + 0.0676·19-s − 1.14·20-s − 4.52·22-s − 23-s − 3.29·25-s + 6.88·26-s − 0.632·28-s + 29-s − 3.80·31-s + 4.53·32-s − 1.70·34-s + 0.939·35-s − 3.80·37-s + 0.0716·38-s − 3.97·40-s + 0.467·41-s + ⋯ |
| L(s) = 1 | + 0.749·2-s − 0.438·4-s + 0.583·5-s + 0.272·7-s − 1.07·8-s + 0.436·10-s − 1.28·11-s + 1.80·13-s + 0.204·14-s − 0.369·16-s − 0.389·17-s + 0.0155·19-s − 0.255·20-s − 0.964·22-s − 0.208·23-s − 0.659·25-s + 1.34·26-s − 0.119·28-s + 0.185·29-s − 0.682·31-s + 0.801·32-s − 0.292·34-s + 0.158·35-s − 0.624·37-s + 0.0116·38-s − 0.628·40-s + 0.0730·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.05T + 2T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 7 | \( 1 - 0.720T + 7T^{2} \) |
| 11 | \( 1 + 4.26T + 11T^{2} \) |
| 13 | \( 1 - 6.49T + 13T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 19 | \( 1 - 0.0676T + 19T^{2} \) |
| 31 | \( 1 + 3.80T + 31T^{2} \) |
| 37 | \( 1 + 3.80T + 37T^{2} \) |
| 41 | \( 1 - 0.467T + 41T^{2} \) |
| 43 | \( 1 - 0.971T + 43T^{2} \) |
| 47 | \( 1 + 7.97T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 4.90T + 59T^{2} \) |
| 61 | \( 1 - 7.29T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 0.843T + 71T^{2} \) |
| 73 | \( 1 + 3.56T + 73T^{2} \) |
| 79 | \( 1 - 3.50T + 79T^{2} \) |
| 83 | \( 1 - 8.90T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 0.847T + 97T^{2} \) |
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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
−8.036048221167308348690597656174, −6.70204514412249048582571332440, −6.14919360156160235323595743359, −5.41496713566803163899786998798, −5.01277948400919531768370425861, −4.00654664682682474916560513151, −3.42541868020511158802717292950, −2.47641730216395080393316494665, −1.45662568297017411635798652251, 0,
1.45662568297017411635798652251, 2.47641730216395080393316494665, 3.42541868020511158802717292950, 4.00654664682682474916560513151, 5.01277948400919531768370425861, 5.41496713566803163899786998798, 6.14919360156160235323595743359, 6.70204514412249048582571332440, 8.036048221167308348690597656174
