L(s) = 1 | + 2.40·2-s − 3-s + 3.78·4-s + 2.51·5-s − 2.40·6-s + 3.82·7-s + 4.29·8-s + 9-s + 6.04·10-s + 5.80·11-s − 3.78·12-s − 13-s + 9.20·14-s − 2.51·15-s + 2.76·16-s + 0.209·17-s + 2.40·18-s − 0.922·19-s + 9.51·20-s − 3.82·21-s + 13.9·22-s − 1.38·23-s − 4.29·24-s + 1.31·25-s − 2.40·26-s − 27-s + 14.4·28-s + ⋯ |
L(s) = 1 | + 1.70·2-s − 0.577·3-s + 1.89·4-s + 1.12·5-s − 0.982·6-s + 1.44·7-s + 1.51·8-s + 0.333·9-s + 1.91·10-s + 1.75·11-s − 1.09·12-s − 0.277·13-s + 2.45·14-s − 0.648·15-s + 0.690·16-s + 0.0507·17-s + 0.566·18-s − 0.211·19-s + 2.12·20-s − 0.834·21-s + 2.97·22-s − 0.288·23-s − 0.877·24-s + 0.263·25-s − 0.471·26-s − 0.192·27-s + 2.73·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.012080864\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.012080864\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 5 | \( 1 - 2.51T + 5T^{2} \) |
| 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 - 5.80T + 11T^{2} \) |
| 17 | \( 1 - 0.209T + 17T^{2} \) |
| 19 | \( 1 + 0.922T + 19T^{2} \) |
| 23 | \( 1 + 1.38T + 23T^{2} \) |
| 29 | \( 1 + 1.48T + 29T^{2} \) |
| 31 | \( 1 - 1.48T + 31T^{2} \) |
| 37 | \( 1 + 0.759T + 37T^{2} \) |
| 41 | \( 1 - 1.09T + 41T^{2} \) |
| 43 | \( 1 + 9.03T + 43T^{2} \) |
| 47 | \( 1 + 5.61T + 47T^{2} \) |
| 53 | \( 1 + 3.43T + 53T^{2} \) |
| 59 | \( 1 + 6.13T + 59T^{2} \) |
| 61 | \( 1 + 5.17T + 61T^{2} \) |
| 67 | \( 1 + 2.42T + 67T^{2} \) |
| 71 | \( 1 + 8.67T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 6.21T + 79T^{2} \) |
| 83 | \( 1 + 0.939T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−8.347832920368910099909262756992, −7.29103217295893927567651400696, −6.57923983375562012067978200190, −6.01884412861898165424737101616, −5.42748534908271080119714834060, −4.65348506839950784054600948119, −4.23701796546955394959466619563, −3.16758785654127038898762609934, −1.87290651739752171538216173818, −1.54792979739044826116696084549,
1.54792979739044826116696084549, 1.87290651739752171538216173818, 3.16758785654127038898762609934, 4.23701796546955394959466619563, 4.65348506839950784054600948119, 5.42748534908271080119714834060, 6.01884412861898165424737101616, 6.57923983375562012067978200190, 7.29103217295893927567651400696, 8.347832920368910099909262756992