| L(s) = 1 | − 0.510·2-s − 3-s − 1.73·4-s + 0.510·6-s + 4.82·7-s + 1.90·8-s + 9-s + 4.88·11-s + 1.73·12-s + 4.59·13-s − 2.46·14-s + 2.50·16-s + 6.50·17-s − 0.510·18-s + 3.09·19-s − 4.82·21-s − 2.49·22-s − 5.62·23-s − 1.90·24-s − 2.34·26-s − 27-s − 8.38·28-s + 29-s + 9.24·31-s − 5.09·32-s − 4.88·33-s − 3.32·34-s + ⋯ |
| L(s) = 1 | − 0.361·2-s − 0.577·3-s − 0.869·4-s + 0.208·6-s + 1.82·7-s + 0.675·8-s + 0.333·9-s + 1.47·11-s + 0.502·12-s + 1.27·13-s − 0.658·14-s + 0.625·16-s + 1.57·17-s − 0.120·18-s + 0.709·19-s − 1.05·21-s − 0.531·22-s − 1.17·23-s − 0.389·24-s − 0.460·26-s − 0.192·27-s − 1.58·28-s + 0.185·29-s + 1.66·31-s − 0.901·32-s − 0.850·33-s − 0.570·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.614060259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.614060259\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.510T + 2T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 - 4.88T + 11T^{2} \) |
| 13 | \( 1 - 4.59T + 13T^{2} \) |
| 17 | \( 1 - 6.50T + 17T^{2} \) |
| 19 | \( 1 - 3.09T + 19T^{2} \) |
| 23 | \( 1 + 5.62T + 23T^{2} \) |
| 31 | \( 1 - 9.24T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 2.84T + 41T^{2} \) |
| 43 | \( 1 + 4.58T + 43T^{2} \) |
| 47 | \( 1 - 3.62T + 47T^{2} \) |
| 53 | \( 1 + 0.967T + 53T^{2} \) |
| 59 | \( 1 + 0.298T + 59T^{2} \) |
| 61 | \( 1 + 0.786T + 61T^{2} \) |
| 67 | \( 1 - 4.86T + 67T^{2} \) |
| 71 | \( 1 - 0.741T + 71T^{2} \) |
| 73 | \( 1 - 5.52T + 73T^{2} \) |
| 79 | \( 1 + 2.96T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 3.67T + 89T^{2} \) |
| 97 | \( 1 + 2.87T + 97T^{2} \) |
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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
−8.920420652755495306587698436555, −8.278796696069314069774650178296, −7.84099132036866466691146849338, −6.74981967358628707902189097715, −5.71565335288668712492348143834, −5.12676978755460206469329976660, −4.23424862164212764585659325676, −3.60937052124579437279199746164, −1.47834099910361017075999504871, −1.14884792375178383344801016190,
1.14884792375178383344801016190, 1.47834099910361017075999504871, 3.60937052124579437279199746164, 4.23424862164212764585659325676, 5.12676978755460206469329976660, 5.71565335288668712492348143834, 6.74981967358628707902189097715, 7.84099132036866466691146849338, 8.278796696069314069774650178296, 8.920420652755495306587698436555
