L(s) = 1 | − 2.81·2-s − 3-s + 5.92·4-s − 4.24·5-s + 2.81·6-s + 0.0516·7-s − 11.0·8-s + 9-s + 11.9·10-s − 4.06·11-s − 5.92·12-s − 4.35·13-s − 0.145·14-s + 4.24·15-s + 19.3·16-s + 17-s − 2.81·18-s − 5.28·19-s − 25.1·20-s − 0.0516·21-s + 11.4·22-s + 6.73·23-s + 11.0·24-s + 13.0·25-s + 12.2·26-s − 27-s + 0.306·28-s + ⋯ |
L(s) = 1 | − 1.99·2-s − 0.577·3-s + 2.96·4-s − 1.89·5-s + 1.14·6-s + 0.0195·7-s − 3.91·8-s + 0.333·9-s + 3.78·10-s − 1.22·11-s − 1.71·12-s − 1.20·13-s − 0.0388·14-s + 1.09·15-s + 4.82·16-s + 0.242·17-s − 0.663·18-s − 1.21·19-s − 5.62·20-s − 0.0112·21-s + 2.43·22-s + 1.40·23-s + 2.25·24-s + 2.60·25-s + 2.40·26-s − 0.192·27-s + 0.0578·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1090792800\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1090792800\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 + 2.81T + 2T^{2} \) |
| 5 | \( 1 + 4.24T + 5T^{2} \) |
| 7 | \( 1 - 0.0516T + 7T^{2} \) |
| 11 | \( 1 + 4.06T + 11T^{2} \) |
| 13 | \( 1 + 4.35T + 13T^{2} \) |
| 19 | \( 1 + 5.28T + 19T^{2} \) |
| 23 | \( 1 - 6.73T + 23T^{2} \) |
| 29 | \( 1 - 6.81T + 29T^{2} \) |
| 31 | \( 1 - 5.25T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 5.37T + 41T^{2} \) |
| 43 | \( 1 - 7.06T + 43T^{2} \) |
| 47 | \( 1 + 6.72T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 + 4.71T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 6.89T + 67T^{2} \) |
| 71 | \( 1 + 3.50T + 71T^{2} \) |
| 73 | \( 1 - 2.03T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 7.14T + 83T^{2} \) |
| 89 | \( 1 - 2.78T + 89T^{2} \) |
| 97 | \( 1 - 8.30T + 97T^{2} \) |
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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
−7.979228912956109715717409041445, −7.37957769555669555437712163136, −6.89202129767789139649175486362, −6.19111350147760556471775370340, −5.04398115949655204712395038811, −4.35378367528105429506412130191, −2.91309275145771411866046622265, −2.73031149175086263759827258201, −1.18856498769353608863706791702, −0.25733497585862303141659854871,
0.25733497585862303141659854871, 1.18856498769353608863706791702, 2.73031149175086263759827258201, 2.91309275145771411866046622265, 4.35378367528105429506412130191, 5.04398115949655204712395038811, 6.19111350147760556471775370340, 6.89202129767789139649175486362, 7.37957769555669555437712163136, 7.979228912956109715717409041445