| L(s) = 1 | − 1.83·2-s + 3-s + 1.38·4-s − 2.07·5-s − 1.83·6-s + 1.13·8-s + 9-s + 3.81·10-s − 6.44·11-s + 1.38·12-s − 0.865·13-s − 2.07·15-s − 4.85·16-s + 2.06·17-s − 1.83·18-s + 3.75·19-s − 2.86·20-s + 11.8·22-s + 5.10·23-s + 1.13·24-s − 0.690·25-s + 1.59·26-s + 27-s + 1.38·29-s + 3.81·30-s − 3.75·31-s + 6.65·32-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.577·3-s + 0.691·4-s − 0.928·5-s − 0.750·6-s + 0.401·8-s + 0.333·9-s + 1.20·10-s − 1.94·11-s + 0.399·12-s − 0.240·13-s − 0.536·15-s − 1.21·16-s + 0.501·17-s − 0.433·18-s + 0.861·19-s − 0.641·20-s + 2.52·22-s + 1.06·23-s + 0.231·24-s − 0.138·25-s + 0.312·26-s + 0.192·27-s + 0.256·29-s + 0.697·30-s − 0.673·31-s + 1.17·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 5 | \( 1 + 2.07T + 5T^{2} \) |
| 11 | \( 1 + 6.44T + 11T^{2} \) |
| 13 | \( 1 + 0.865T + 13T^{2} \) |
| 17 | \( 1 - 2.06T + 17T^{2} \) |
| 19 | \( 1 - 3.75T + 19T^{2} \) |
| 23 | \( 1 - 5.10T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 3.75T + 31T^{2} \) |
| 37 | \( 1 + 0.654T + 37T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 0.611T + 53T^{2} \) |
| 59 | \( 1 - 3.36T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 3.50T + 71T^{2} \) |
| 73 | \( 1 + 0.660T + 73T^{2} \) |
| 79 | \( 1 + 3.56T + 79T^{2} \) |
| 83 | \( 1 - 7.34T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−7.74636151255863891963353882461, −7.54850965472164661586192037670, −6.81037084262389615761758987680, −5.42379944619930553967793506413, −4.90360679054563413979273088041, −3.89035391865833838862395012478, −2.99964075585920627304110273217, −2.26868155113948014389610823818, −1.01819833907776958802554762406, 0,
1.01819833907776958802554762406, 2.26868155113948014389610823818, 2.99964075585920627304110273217, 3.89035391865833838862395012478, 4.90360679054563413979273088041, 5.42379944619930553967793506413, 6.81037084262389615761758987680, 7.54850965472164661586192037670, 7.74636151255863891963353882461
