L(s) = 1 | + 0.343·2-s − 2.89·3-s − 1.88·4-s + 0.224·5-s − 0.992·6-s + 0.410·7-s − 1.33·8-s + 5.35·9-s + 0.0769·10-s + 2.20·11-s + 5.44·12-s − 13-s + 0.141·14-s − 0.648·15-s + 3.30·16-s + 3.43·17-s + 1.83·18-s − 4.36·19-s − 0.421·20-s − 1.18·21-s + 0.756·22-s + 7.24·23-s + 3.85·24-s − 4.94·25-s − 0.343·26-s − 6.82·27-s − 0.773·28-s + ⋯ |
L(s) = 1 | + 0.242·2-s − 1.66·3-s − 0.941·4-s + 0.100·5-s − 0.405·6-s + 0.155·7-s − 0.471·8-s + 1.78·9-s + 0.0243·10-s + 0.664·11-s + 1.57·12-s − 0.277·13-s + 0.0376·14-s − 0.167·15-s + 0.826·16-s + 0.834·17-s + 0.433·18-s − 1.00·19-s − 0.0943·20-s − 0.259·21-s + 0.161·22-s + 1.51·23-s + 0.786·24-s − 0.989·25-s − 0.0673·26-s − 1.31·27-s − 0.146·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 - 0.343T + 2T^{2} \) |
| 3 | \( 1 + 2.89T + 3T^{2} \) |
| 5 | \( 1 - 0.224T + 5T^{2} \) |
| 7 | \( 1 - 0.410T + 7T^{2} \) |
| 11 | \( 1 - 2.20T + 11T^{2} \) |
| 17 | \( 1 - 3.43T + 17T^{2} \) |
| 19 | \( 1 + 4.36T + 19T^{2} \) |
| 23 | \( 1 - 7.24T + 23T^{2} \) |
| 29 | \( 1 + 4.98T + 29T^{2} \) |
| 31 | \( 1 - 3.02T + 31T^{2} \) |
| 37 | \( 1 + 2.20T + 37T^{2} \) |
| 41 | \( 1 + 8.50T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 9.16T + 59T^{2} \) |
| 61 | \( 1 - 2.85T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 0.618T + 73T^{2} \) |
| 83 | \( 1 + 0.151T + 83T^{2} \) |
| 89 | \( 1 + 4.84T + 89T^{2} \) |
| 97 | \( 1 + 7.53T + 97T^{2} \) |
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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
−9.694549658559830491346184687961, −8.861514493340362535190709281844, −7.74434876939033590030526111880, −6.67691233114838237192777170031, −5.95258515686188618097894048833, −5.14210381067622801069189046087, −4.55508372133206573864592257536, −3.49439086481661296767939936926, −1.39139036063280011552091756993, 0,
1.39139036063280011552091756993, 3.49439086481661296767939936926, 4.55508372133206573864592257536, 5.14210381067622801069189046087, 5.95258515686188618097894048833, 6.67691233114838237192777170031, 7.74434876939033590030526111880, 8.861514493340362535190709281844, 9.694549658559830491346184687961