L(s) = 1 | − 10.2·3-s + 3.08·5-s − 7.31·7-s + 77.6·9-s − 23.9·11-s − 35.5·13-s − 31.5·15-s + 17·17-s + 34.2·19-s + 74.7·21-s − 149.·23-s − 115.·25-s − 517.·27-s + 120.·29-s + 247.·31-s + 244.·33-s − 22.5·35-s + 448.·37-s + 363.·39-s + 303.·41-s + 194.·43-s + 239.·45-s + 21.0·47-s − 289.·49-s − 173.·51-s + 362.·53-s − 73.8·55-s + ⋯ |
L(s) = 1 | − 1.96·3-s + 0.275·5-s − 0.394·7-s + 2.87·9-s − 0.656·11-s − 0.758·13-s − 0.542·15-s + 0.242·17-s + 0.413·19-s + 0.777·21-s − 1.35·23-s − 0.923·25-s − 3.69·27-s + 0.769·29-s + 1.43·31-s + 1.29·33-s − 0.108·35-s + 1.99·37-s + 1.49·39-s + 1.15·41-s + 0.688·43-s + 0.792·45-s + 0.0654·47-s − 0.844·49-s − 0.477·51-s + 0.939·53-s − 0.180·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - 17T \) |
good | 3 | \( 1 + 10.2T + 27T^{2} \) |
| 5 | \( 1 - 3.08T + 125T^{2} \) |
| 7 | \( 1 + 7.31T + 343T^{2} \) |
| 11 | \( 1 + 23.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 35.5T + 2.19e3T^{2} \) |
| 19 | \( 1 - 34.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 120.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 247.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 448.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 303.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 194.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 21.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 362.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 364.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 478.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 5.17T + 3.00e5T^{2} \) |
| 71 | \( 1 - 335.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.08e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 561.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 746.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 247.T + 9.12e5T^{2} \) |
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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
−9.636943915777853806593842058850, −7.925750763887971701159396890102, −7.27017091106400434376444247531, −6.12818818036513343144514201575, −5.91095626916583268644978375058, −4.85792758079194603747574098647, −4.16048911855976402078379384778, −2.45869618422995885067054643223, −1.01035587929278917691241898120, 0,
1.01035587929278917691241898120, 2.45869618422995885067054643223, 4.16048911855976402078379384778, 4.85792758079194603747574098647, 5.91095626916583268644978375058, 6.12818818036513343144514201575, 7.27017091106400434376444247531, 7.925750763887971701159396890102, 9.636943915777853806593842058850