| L(s) = 1 | − 2-s − 3-s + 4-s + 1.54·5-s + 6-s + 2.97·7-s − 8-s + 9-s − 1.54·10-s − 1.81·11-s − 12-s − 2.96·13-s − 2.97·14-s − 1.54·15-s + 16-s − 17-s − 18-s + 0.429·19-s + 1.54·20-s − 2.97·21-s + 1.81·22-s + 4.24·23-s + 24-s − 2.62·25-s + 2.96·26-s − 27-s + 2.97·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.689·5-s + 0.408·6-s + 1.12·7-s − 0.353·8-s + 0.333·9-s − 0.487·10-s − 0.546·11-s − 0.288·12-s − 0.823·13-s − 0.793·14-s − 0.398·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.0985·19-s + 0.344·20-s − 0.648·21-s + 0.386·22-s + 0.886·23-s + 0.204·24-s − 0.524·25-s + 0.582·26-s − 0.192·27-s + 0.561·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 1.54T + 5T^{2} \) |
| 7 | \( 1 - 2.97T + 7T^{2} \) |
| 11 | \( 1 + 1.81T + 11T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 19 | \( 1 - 0.429T + 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 + 1.25T + 29T^{2} \) |
| 31 | \( 1 + 0.520T + 31T^{2} \) |
| 37 | \( 1 + 2.23T + 37T^{2} \) |
| 41 | \( 1 + 7.24T + 41T^{2} \) |
| 43 | \( 1 + 4.26T + 43T^{2} \) |
| 47 | \( 1 - 5.31T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 6.97T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 4.18T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 + 3.10T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 + 1.78T + 97T^{2} \) |
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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
−7.62945598973356288435512504005, −7.19920197246137207398077866190, −6.36589046146685785121739903331, −5.46445194997356677100575495407, −5.11736635521427048879733887751, −4.23604865431563947809000573126, −2.92417586728736967896063380642, −2.04218422346585162635887690618, −1.34445680228574244214733624357, 0,
1.34445680228574244214733624357, 2.04218422346585162635887690618, 2.92417586728736967896063380642, 4.23604865431563947809000573126, 5.11736635521427048879733887751, 5.46445194997356677100575495407, 6.36589046146685785121739903331, 7.19920197246137207398077866190, 7.62945598973356288435512504005
