| L(s) = 1 | + 2.59·2-s − 3-s + 4.73·4-s + 2.19·5-s − 2.59·6-s + 7.08·8-s + 9-s + 5.68·10-s + 11-s − 4.73·12-s − 2.95·13-s − 2.19·15-s + 8.92·16-s + 2.59·17-s + 2.59·18-s − 2.35·19-s + 10.3·20-s + 2.59·22-s + 8.92·23-s − 7.08·24-s − 0.195·25-s − 7.66·26-s − 27-s + 4.48·29-s − 5.68·30-s − 1.73·31-s + 8.97·32-s + ⋯ |
| L(s) = 1 | + 1.83·2-s − 0.577·3-s + 2.36·4-s + 0.980·5-s − 1.05·6-s + 2.50·8-s + 0.333·9-s + 1.79·10-s + 0.301·11-s − 1.36·12-s − 0.818·13-s − 0.565·15-s + 2.23·16-s + 0.629·17-s + 0.611·18-s − 0.541·19-s + 2.31·20-s + 0.553·22-s + 1.86·23-s − 1.44·24-s − 0.0390·25-s − 1.50·26-s − 0.192·27-s + 0.833·29-s − 1.03·30-s − 0.310·31-s + 1.58·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.316922114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.316922114\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.59T + 2T^{2} \) |
| 5 | \( 1 - 2.19T + 5T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 - 2.59T + 17T^{2} \) |
| 19 | \( 1 + 2.35T + 19T^{2} \) |
| 23 | \( 1 - 8.92T + 23T^{2} \) |
| 29 | \( 1 - 4.48T + 29T^{2} \) |
| 31 | \( 1 + 1.73T + 31T^{2} \) |
| 37 | \( 1 + 4.38T + 37T^{2} \) |
| 41 | \( 1 + 9.68T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 9.17T + 53T^{2} \) |
| 59 | \( 1 - 8.92T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 3.25T + 67T^{2} \) |
| 71 | \( 1 - 0.994T + 71T^{2} \) |
| 73 | \( 1 - 0.294T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 6.26T + 83T^{2} \) |
| 89 | \( 1 + 2.60T + 89T^{2} \) |
| 97 | \( 1 - 7.51T + 97T^{2} \) |
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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
−9.754770410152948315410543457496, −8.533971559678773459223708668876, −7.08504328393626721635070395616, −6.80618205530517578495139778586, −5.84109411133430966953657024122, −5.23925158030536799041506823920, −4.67766892596166993209710793301, −3.53936292985319029795588198803, −2.58755663370544267455734783220, −1.54166932842709806120021572676,
1.54166932842709806120021572676, 2.58755663370544267455734783220, 3.53936292985319029795588198803, 4.67766892596166993209710793301, 5.23925158030536799041506823920, 5.84109411133430966953657024122, 6.80618205530517578495139778586, 7.08504328393626721635070395616, 8.533971559678773459223708668876, 9.754770410152948315410543457496
