| L(s) = 1 | − 2.68·2-s + 3-s + 5.21·4-s + 5-s − 2.68·6-s + 4.21·7-s − 8.64·8-s + 9-s − 2.68·10-s + 5.96·11-s + 5.21·12-s − 3.61·13-s − 11.3·14-s + 15-s + 12.7·16-s + 2.60·17-s − 2.68·18-s − 3.75·19-s + 5.21·20-s + 4.21·21-s − 16.0·22-s − 8.64·24-s + 25-s + 9.71·26-s + 27-s + 22.0·28-s + 10.1·29-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 0.577·3-s + 2.60·4-s + 0.447·5-s − 1.09·6-s + 1.59·7-s − 3.05·8-s + 0.333·9-s − 0.849·10-s + 1.79·11-s + 1.50·12-s − 1.00·13-s − 3.02·14-s + 0.258·15-s + 3.19·16-s + 0.631·17-s − 0.633·18-s − 0.861·19-s + 1.16·20-s + 0.920·21-s − 3.41·22-s − 1.76·24-s + 0.200·25-s + 1.90·26-s + 0.192·27-s + 4.16·28-s + 1.89·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.736597874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.736597874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 7 | \( 1 - 4.21T + 7T^{2} \) |
| 11 | \( 1 - 5.96T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 + 3.75T + 19T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 2.21T + 31T^{2} \) |
| 37 | \( 1 + 8.73T + 37T^{2} \) |
| 41 | \( 1 + 2.60T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 + 1.74T + 53T^{2} \) |
| 59 | \( 1 - 8.03T + 59T^{2} \) |
| 61 | \( 1 + 1.61T + 61T^{2} \) |
| 67 | \( 1 + 1.04T + 67T^{2} \) |
| 71 | \( 1 + 6.25T + 71T^{2} \) |
| 73 | \( 1 - 7.13T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 0.127T + 89T^{2} \) |
| 97 | \( 1 + 0.233T + 97T^{2} \) |
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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
−8.214835442976816177074234358734, −7.31904170955628556370575275879, −6.83427694276598656014745764031, −6.18943041070555030117084414075, −5.09339086832931415040690358278, −4.25235976781918941316637211070, −3.08520773450335091924265146835, −2.14418221440057435951440742831, −1.62374687721648749677828423893, −0.920617177770825399283069423722,
0.920617177770825399283069423722, 1.62374687721648749677828423893, 2.14418221440057435951440742831, 3.08520773450335091924265146835, 4.25235976781918941316637211070, 5.09339086832931415040690358278, 6.18943041070555030117084414075, 6.83427694276598656014745764031, 7.31904170955628556370575275879, 8.214835442976816177074234358734
