| L(s) = 1 | − 0.833·2-s + 3-s − 1.30·4-s − 0.833·6-s + 1.65·7-s + 2.75·8-s + 9-s + 5.28·11-s − 1.30·12-s + 0.534·13-s − 1.38·14-s + 0.311·16-s + 7.84·17-s − 0.833·18-s + 3.07·19-s + 1.65·21-s − 4.40·22-s − 5.72·23-s + 2.75·24-s − 0.445·26-s + 27-s − 2.16·28-s − 2.72·29-s + 6.36·31-s − 5.77·32-s + 5.28·33-s − 6.54·34-s + ⋯ |
| L(s) = 1 | − 0.589·2-s + 0.577·3-s − 0.652·4-s − 0.340·6-s + 0.626·7-s + 0.974·8-s + 0.333·9-s + 1.59·11-s − 0.376·12-s + 0.148·13-s − 0.369·14-s + 0.0778·16-s + 1.90·17-s − 0.196·18-s + 0.706·19-s + 0.361·21-s − 0.939·22-s − 1.19·23-s + 0.562·24-s − 0.0873·26-s + 0.192·27-s − 0.408·28-s − 0.506·29-s + 1.14·31-s − 1.02·32-s + 0.919·33-s − 1.12·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.992542076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.992542076\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 0.833T + 2T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 - 5.28T + 11T^{2} \) |
| 13 | \( 1 - 0.534T + 13T^{2} \) |
| 17 | \( 1 - 7.84T + 17T^{2} \) |
| 19 | \( 1 - 3.07T + 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 29 | \( 1 + 2.72T + 29T^{2} \) |
| 31 | \( 1 - 6.36T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 1.07T + 41T^{2} \) |
| 43 | \( 1 - 0.928T + 43T^{2} \) |
| 53 | \( 1 + 7.37T + 53T^{2} \) |
| 59 | \( 1 - 5.63T + 59T^{2} \) |
| 61 | \( 1 + 1.87T + 61T^{2} \) |
| 67 | \( 1 + 4.20T + 67T^{2} \) |
| 71 | \( 1 + 16.4T + 71T^{2} \) |
| 73 | \( 1 + 7.74T + 73T^{2} \) |
| 79 | \( 1 + 7.48T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 1.59T + 89T^{2} \) |
| 97 | \( 1 - 5.08T + 97T^{2} \) |
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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
−8.590552137171256156808632441327, −7.79757869376113304624076516690, −7.62877371283031281059954723018, −6.37453731533303297322054220701, −5.57825477032242766686569905782, −4.53884212398585695518139831333, −3.96328803252630598891626327389, −3.10332589082426884435316340761, −1.61079860913491625422365128680, −1.03003324742684792537249720519,
1.03003324742684792537249720519, 1.61079860913491625422365128680, 3.10332589082426884435316340761, 3.96328803252630598891626327389, 4.53884212398585695518139831333, 5.57825477032242766686569905782, 6.37453731533303297322054220701, 7.62877371283031281059954723018, 7.79757869376113304624076516690, 8.590552137171256156808632441327
