| L(s) = 1 | + 2-s − 0.352·3-s + 4-s − 1.51·5-s − 0.352·6-s − 1.04·7-s + 8-s − 2.87·9-s − 1.51·10-s + 3.44·11-s − 0.352·12-s − 5.23·13-s − 1.04·14-s + 0.533·15-s + 16-s − 0.458·17-s − 2.87·18-s + 0.0118·19-s − 1.51·20-s + 0.366·21-s + 3.44·22-s + 23-s − 0.352·24-s − 2.70·25-s − 5.23·26-s + 2.07·27-s − 1.04·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.203·3-s + 0.5·4-s − 0.677·5-s − 0.143·6-s − 0.393·7-s + 0.353·8-s − 0.958·9-s − 0.479·10-s + 1.03·11-s − 0.101·12-s − 1.45·13-s − 0.278·14-s + 0.137·15-s + 0.250·16-s − 0.111·17-s − 0.677·18-s + 0.00271·19-s − 0.338·20-s + 0.0800·21-s + 0.733·22-s + 0.208·23-s − 0.0719·24-s − 0.540·25-s − 1.02·26-s + 0.398·27-s − 0.196·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.783751298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.783751298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 0.352T + 3T^{2} \) |
| 5 | \( 1 + 1.51T + 5T^{2} \) |
| 7 | \( 1 + 1.04T + 7T^{2} \) |
| 11 | \( 1 - 3.44T + 11T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 + 0.458T + 17T^{2} \) |
| 19 | \( 1 - 0.0118T + 19T^{2} \) |
| 29 | \( 1 - 5.71T + 29T^{2} \) |
| 31 | \( 1 + 7.95T + 31T^{2} \) |
| 37 | \( 1 - 3.61T + 37T^{2} \) |
| 41 | \( 1 + 4.81T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 8.16T + 47T^{2} \) |
| 53 | \( 1 - 0.539T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 2.65T + 61T^{2} \) |
| 67 | \( 1 - 7.91T + 67T^{2} \) |
| 71 | \( 1 + 2.66T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 3.78T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 1.83T + 89T^{2} \) |
| 97 | \( 1 + 7.88T + 97T^{2} \) |
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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
−7.892622376097437218640089462348, −7.23788555187353427577286750171, −6.63817759730654136733638925189, −5.86734032607925117104198580352, −5.20721026093709192049703891074, −4.38160117900343645649948344812, −3.73562769867504975049677216345, −2.91646221063586706325048612258, −2.12152695260340553585301137123, −0.60915918656328628459099410945,
0.60915918656328628459099410945, 2.12152695260340553585301137123, 2.91646221063586706325048612258, 3.73562769867504975049677216345, 4.38160117900343645649948344812, 5.20721026093709192049703891074, 5.86734032607925117104198580352, 6.63817759730654136733638925189, 7.23788555187353427577286750171, 7.892622376097437218640089462348
