| L(s) = 1 | − 2-s + 3-s + 4-s + 2.77·5-s − 6-s − 0.160·7-s − 8-s + 9-s − 2.77·10-s + 5.18·11-s + 12-s + 0.0641·13-s + 0.160·14-s + 2.77·15-s + 16-s + 7.56·17-s − 18-s − 19-s + 2.77·20-s − 0.160·21-s − 5.18·22-s + 8.87·23-s − 24-s + 2.67·25-s − 0.0641·26-s + 27-s − 0.160·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.23·5-s − 0.408·6-s − 0.0605·7-s − 0.353·8-s + 0.333·9-s − 0.876·10-s + 1.56·11-s + 0.288·12-s + 0.0177·13-s + 0.0428·14-s + 0.715·15-s + 0.250·16-s + 1.83·17-s − 0.235·18-s − 0.229·19-s + 0.619·20-s − 0.0349·21-s − 1.10·22-s + 1.85·23-s − 0.204·24-s + 0.535·25-s − 0.0125·26-s + 0.192·27-s − 0.0302·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.040246309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.040246309\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 - 2.77T + 5T^{2} \) |
| 7 | \( 1 + 0.160T + 7T^{2} \) |
| 11 | \( 1 - 5.18T + 11T^{2} \) |
| 13 | \( 1 - 0.0641T + 13T^{2} \) |
| 17 | \( 1 - 7.56T + 17T^{2} \) |
| 23 | \( 1 - 8.87T + 23T^{2} \) |
| 29 | \( 1 - 7.27T + 29T^{2} \) |
| 31 | \( 1 - 4.20T + 31T^{2} \) |
| 37 | \( 1 + 7.18T + 37T^{2} \) |
| 41 | \( 1 - 2.41T + 41T^{2} \) |
| 43 | \( 1 + 2.58T + 43T^{2} \) |
| 47 | \( 1 - 3.47T + 47T^{2} \) |
| 59 | \( 1 + 7.70T + 59T^{2} \) |
| 61 | \( 1 + 3.26T + 61T^{2} \) |
| 67 | \( 1 + 3.79T + 67T^{2} \) |
| 71 | \( 1 + 3.48T + 71T^{2} \) |
| 73 | \( 1 + 1.80T + 73T^{2} \) |
| 79 | \( 1 + 7.76T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 1.01T + 97T^{2} \) |
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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
−8.320295801338500971898624959914, −7.32871063207823048980834515669, −6.74780983896807519280140885870, −6.13096313516843666217162930390, −5.37324823482250573546904809768, −4.41930721036580720710502358826, −3.27656286678390848471973189811, −2.78533508606323705306474226731, −1.48768202244720110125362376606, −1.21090718255665752319530673002,
1.21090718255665752319530673002, 1.48768202244720110125362376606, 2.78533508606323705306474226731, 3.27656286678390848471973189811, 4.41930721036580720710502358826, 5.37324823482250573546904809768, 6.13096313516843666217162930390, 6.74780983896807519280140885870, 7.32871063207823048980834515669, 8.320295801338500971898624959914
