| L(s) = 1 | − 0.494·2-s − 1.75·4-s + 1.77·5-s + 3.57·7-s + 1.85·8-s − 0.879·10-s + 2.98·11-s − 1.32·13-s − 1.76·14-s + 2.59·16-s + 3.01·19-s − 3.12·20-s − 1.47·22-s − 6.89·23-s − 1.83·25-s + 0.657·26-s − 6.26·28-s + 9.41·29-s − 1.84·31-s − 4.99·32-s + 6.35·35-s + 10.8·37-s − 1.49·38-s + 3.30·40-s + 10.1·41-s − 1.79·43-s − 5.24·44-s + ⋯ |
| L(s) = 1 | − 0.349·2-s − 0.877·4-s + 0.795·5-s + 1.34·7-s + 0.656·8-s − 0.278·10-s + 0.900·11-s − 0.368·13-s − 0.471·14-s + 0.648·16-s + 0.692·19-s − 0.698·20-s − 0.314·22-s − 1.43·23-s − 0.366·25-s + 0.128·26-s − 1.18·28-s + 1.74·29-s − 0.330·31-s − 0.883·32-s + 1.07·35-s + 1.77·37-s − 0.242·38-s + 0.522·40-s + 1.58·41-s − 0.273·43-s − 0.790·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.172459350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.172459350\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.494T + 2T^{2} \) |
| 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 13 | \( 1 + 1.32T + 13T^{2} \) |
| 19 | \( 1 - 3.01T + 19T^{2} \) |
| 23 | \( 1 + 6.89T + 23T^{2} \) |
| 29 | \( 1 - 9.41T + 29T^{2} \) |
| 31 | \( 1 + 1.84T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 1.79T + 43T^{2} \) |
| 47 | \( 1 - 0.646T + 47T^{2} \) |
| 53 | \( 1 + 2.51T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 7.75T + 61T^{2} \) |
| 67 | \( 1 + 8.74T + 67T^{2} \) |
| 71 | \( 1 - 1.90T + 71T^{2} \) |
| 73 | \( 1 + 2.95T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 - 3.83T + 89T^{2} \) |
| 97 | \( 1 - 5.52T + 97T^{2} \) |
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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
−7.80163813188376950067299447136, −7.56360787881755736271878160089, −6.24781655853175654276082926900, −5.85712435329158977205109224159, −4.83735280800898268784880611062, −4.53691656256615857999722463950, −3.69448809272673141537102030309, −2.43845235647555638036026496989, −1.59771236527726032787245165059, −0.855734307296194371988869652679,
0.855734307296194371988869652679, 1.59771236527726032787245165059, 2.43845235647555638036026496989, 3.69448809272673141537102030309, 4.53691656256615857999722463950, 4.83735280800898268784880611062, 5.85712435329158977205109224159, 6.24781655853175654276082926900, 7.56360787881755736271878160089, 7.80163813188376950067299447136
