| L(s) = 1 | − 1.16i·2-s + 0.645·4-s + 2.64·7-s − 3.07i·8-s − 6.57i·11-s − 3.07i·14-s − 2.29·16-s − 7.64·22-s − 1.91i·23-s + 1.70·28-s + 8.89i·29-s − 3.49i·32-s − 10.5·37-s + 5.29·43-s − 4.24i·44-s + ⋯ |
| L(s) = 1 | − 0.822i·2-s + 0.322·4-s + 0.999·7-s − 1.08i·8-s − 1.98i·11-s − 0.822i·14-s − 0.572·16-s − 1.63·22-s − 0.399i·23-s + 0.322·28-s + 1.65i·29-s − 0.617i·32-s − 1.73·37-s + 0.806·43-s − 0.639i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.131978389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131978389\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 2 | \( 1 + 1.16iT - 2T^{2} \) |
| 11 | \( 1 + 6.57iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 1.91iT - 23T^{2} \) |
| 29 | \( 1 - 8.89iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 0.412iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 15.0iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.044959443311421457180471414542, −8.547767274993075087297279725831, −7.63167604965816847753459454626, −6.71595373065199350240138613809, −5.83159702892474670142972799150, −4.95324942122393747081156992276, −3.71174307494864839037167559787, −3.05769001791890848389834561012, −1.88212149471879291596206431913, −0.830088934259370455168746585630,
1.71436586487546910800778629431, 2.42209821186719376666119312660, 4.09724138510205194607742399311, 4.90611365990112501245130557397, 5.60678939072480557305574146205, 6.67187287926856091829365227287, 7.34499818166505059255869613852, 7.84417318866743676699972446377, 8.671519541878905881471133976238, 9.666403805550574654493044551269
