| L(s) = 1 | + 2.37·2-s − 0.630·3-s + 3.62·4-s − 1.61·5-s − 1.49·6-s + 3.60·7-s + 3.85·8-s − 2.60·9-s − 3.83·10-s + 3.25·11-s − 2.28·12-s + 6.29·13-s + 8.55·14-s + 1.01·15-s + 1.88·16-s − 2.89·17-s − 6.17·18-s − 2.32·19-s − 5.85·20-s − 2.27·21-s + 7.72·22-s + 2.72·23-s − 2.43·24-s − 2.39·25-s + 14.9·26-s + 3.53·27-s + 13.0·28-s + ⋯ |
| L(s) = 1 | + 1.67·2-s − 0.364·3-s + 1.81·4-s − 0.722·5-s − 0.610·6-s + 1.36·7-s + 1.36·8-s − 0.867·9-s − 1.21·10-s + 0.981·11-s − 0.660·12-s + 1.74·13-s + 2.28·14-s + 0.263·15-s + 0.471·16-s − 0.702·17-s − 1.45·18-s − 0.533·19-s − 1.30·20-s − 0.496·21-s + 1.64·22-s + 0.567·23-s − 0.496·24-s − 0.478·25-s + 2.92·26-s + 0.680·27-s + 2.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.388755472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.388755472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 3 | \( 1 + 0.630T + 3T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 - 6.29T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 + 2.32T + 19T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 - 7.05T + 29T^{2} \) |
| 31 | \( 1 - 5.28T + 31T^{2} \) |
| 37 | \( 1 + 0.106T + 37T^{2} \) |
| 41 | \( 1 - 8.84T + 41T^{2} \) |
| 47 | \( 1 + 0.939T + 47T^{2} \) |
| 53 | \( 1 - 2.33T + 53T^{2} \) |
| 59 | \( 1 - 6.07T + 59T^{2} \) |
| 61 | \( 1 + 5.89T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 5.21T + 71T^{2} \) |
| 73 | \( 1 + 4.61T + 73T^{2} \) |
| 79 | \( 1 + 1.77T + 79T^{2} \) |
| 83 | \( 1 - 6.67T + 83T^{2} \) |
| 89 | \( 1 + 2.38T + 89T^{2} \) |
| 97 | \( 1 - 7.44T + 97T^{2} \) |
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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
−8.832783346398677375721828460225, −8.488183432469526625852106556604, −7.47157255900485465844852488871, −6.31088145933957749738980574017, −6.10234203596092848498545435315, −4.95329812121069442266239005058, −4.35035176600187868643361356433, −3.71121718798918604835486405984, −2.61768883904971704799646377808, −1.27637748749192059215245561097,
1.27637748749192059215245561097, 2.61768883904971704799646377808, 3.71121718798918604835486405984, 4.35035176600187868643361356433, 4.95329812121069442266239005058, 6.10234203596092848498545435315, 6.31088145933957749738980574017, 7.47157255900485465844852488871, 8.488183432469526625852106556604, 8.832783346398677375721828460225
