| L(s) = 1 | − 1.88·2-s + 1.31·3-s + 1.55·4-s − 5-s − 2.48·6-s + 0.927·7-s + 0.831·8-s − 1.26·9-s + 1.88·10-s + 11-s + 2.05·12-s − 3.54·13-s − 1.74·14-s − 1.31·15-s − 4.68·16-s − 0.175·17-s + 2.38·18-s + 19-s − 1.55·20-s + 1.22·21-s − 1.88·22-s − 3.79·23-s + 1.09·24-s + 25-s + 6.69·26-s − 5.61·27-s + 1.44·28-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.760·3-s + 0.779·4-s − 0.447·5-s − 1.01·6-s + 0.350·7-s + 0.293·8-s − 0.420·9-s + 0.596·10-s + 0.301·11-s + 0.593·12-s − 0.984·13-s − 0.467·14-s − 0.340·15-s − 1.17·16-s − 0.0425·17-s + 0.561·18-s + 0.229·19-s − 0.348·20-s + 0.266·21-s − 0.402·22-s − 0.791·23-s + 0.223·24-s + 0.200·25-s + 1.31·26-s − 1.08·27-s + 0.273·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.88T + 2T^{2} \) |
| 3 | \( 1 - 1.31T + 3T^{2} \) |
| 7 | \( 1 - 0.927T + 7T^{2} \) |
| 13 | \( 1 + 3.54T + 13T^{2} \) |
| 17 | \( 1 + 0.175T + 17T^{2} \) |
| 23 | \( 1 + 3.79T + 23T^{2} \) |
| 29 | \( 1 + 8.73T + 29T^{2} \) |
| 31 | \( 1 - 9.67T + 31T^{2} \) |
| 37 | \( 1 + 0.0348T + 37T^{2} \) |
| 41 | \( 1 + 4.58T + 41T^{2} \) |
| 43 | \( 1 + 5.05T + 43T^{2} \) |
| 47 | \( 1 - 8.40T + 47T^{2} \) |
| 53 | \( 1 + 3.94T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 0.124T + 71T^{2} \) |
| 73 | \( 1 - 6.73T + 73T^{2} \) |
| 79 | \( 1 + 1.90T + 79T^{2} \) |
| 83 | \( 1 + 3.01T + 83T^{2} \) |
| 89 | \( 1 + 7.37T + 89T^{2} \) |
| 97 | \( 1 + 3.87T + 97T^{2} \) |
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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
−9.426682724357321219318944987954, −8.669340535164036888500673754141, −7.938592802572712893912530834110, −7.56616099807254388193709865816, −6.47115624425926123304451193375, −5.09639058311708306122842376696, −4.03746707615882169194304530119, −2.78320595906033644870753637763, −1.69302173902050099153784244115, 0,
1.69302173902050099153784244115, 2.78320595906033644870753637763, 4.03746707615882169194304530119, 5.09639058311708306122842376696, 6.47115624425926123304451193375, 7.56616099807254388193709865816, 7.938592802572712893912530834110, 8.669340535164036888500673754141, 9.426682724357321219318944987954
