| L(s) = 1 | + 0.618·2-s − 1.61·4-s + 3·7-s − 2.23·8-s + 5.47·11-s + 6.23·13-s + 1.85·14-s + 1.85·16-s + 3.47·17-s + 7.70·19-s + 3.38·22-s + 3.85·26-s − 4.85·28-s − 29-s − 8·31-s + 5.61·32-s + 2.14·34-s + 8·37-s + 4.76·38-s + 4.47·41-s − 3.23·43-s − 8.85·44-s + 6.70·47-s + 2·49-s − 10.0·52-s − 6.76·53-s − 6.70·56-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.809·4-s + 1.13·7-s − 0.790·8-s + 1.64·11-s + 1.72·13-s + 0.495·14-s + 0.463·16-s + 0.842·17-s + 1.76·19-s + 0.721·22-s + 0.755·26-s − 0.917·28-s − 0.185·29-s − 1.43·31-s + 0.993·32-s + 0.368·34-s + 1.31·37-s + 0.772·38-s + 0.698·41-s − 0.493·43-s − 1.33·44-s + 0.978·47-s + 0.285·49-s − 1.39·52-s − 0.929·53-s − 0.896·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.264605918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.264605918\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 5.47T + 11T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 - 3.47T + 17T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 3.23T + 43T^{2} \) |
| 47 | \( 1 - 6.70T + 47T^{2} \) |
| 53 | \( 1 + 6.76T + 53T^{2} \) |
| 59 | \( 1 + 5.23T + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 7.23T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + 6.18T + 79T^{2} \) |
| 83 | \( 1 + 3.70T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 2.76T + 97T^{2} \) |
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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
−8.040406429657753405828417545233, −7.43527198631723347953430121433, −6.38422253454370398577104981968, −5.72899459123284017770723157425, −5.21770040889114517555638204420, −4.23795454770776330975962457280, −3.80058502672385595447266020499, −3.10361143848694433688060143986, −1.43888579542215107097389660009, −1.08636406122801098652270404839,
1.08636406122801098652270404839, 1.43888579542215107097389660009, 3.10361143848694433688060143986, 3.80058502672385595447266020499, 4.23795454770776330975962457280, 5.21770040889114517555638204420, 5.72899459123284017770723157425, 6.38422253454370398577104981968, 7.43527198631723347953430121433, 8.040406429657753405828417545233
