| L(s) = 1 | − 1.66·2-s + 3.41·3-s + 0.765·4-s − 1.23·5-s − 5.68·6-s + 2.05·8-s + 8.68·9-s + 2.05·10-s + 11-s + 2.61·12-s + 13-s − 4.23·15-s − 4.94·16-s − 4.94·17-s − 14.4·18-s + 1.61·19-s − 0.946·20-s − 1.66·22-s − 3.36·23-s + 7.02·24-s − 3.46·25-s − 1.66·26-s + 19.4·27-s + 5.05·29-s + 7.03·30-s + 7.91·31-s + 4.11·32-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 1.97·3-s + 0.382·4-s − 0.553·5-s − 2.32·6-s + 0.726·8-s + 2.89·9-s + 0.650·10-s + 0.301·11-s + 0.755·12-s + 0.277·13-s − 1.09·15-s − 1.23·16-s − 1.19·17-s − 3.40·18-s + 0.371·19-s − 0.211·20-s − 0.354·22-s − 0.701·23-s + 1.43·24-s − 0.693·25-s − 0.326·26-s + 3.74·27-s + 0.938·29-s + 1.28·30-s + 1.42·31-s + 0.727·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.163197279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.163197279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.66T + 2T^{2} \) |
| 3 | \( 1 - 3.41T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 17 | \( 1 + 4.94T + 17T^{2} \) |
| 19 | \( 1 - 1.61T + 19T^{2} \) |
| 23 | \( 1 + 3.36T + 23T^{2} \) |
| 29 | \( 1 - 5.05T + 29T^{2} \) |
| 31 | \( 1 - 7.91T + 31T^{2} \) |
| 37 | \( 1 - 1.91T + 37T^{2} \) |
| 41 | \( 1 - 1.43T + 41T^{2} \) |
| 43 | \( 1 - 9.12T + 43T^{2} \) |
| 47 | \( 1 + 9.56T + 47T^{2} \) |
| 53 | \( 1 - 6.84T + 53T^{2} \) |
| 59 | \( 1 + 5.36T + 59T^{2} \) |
| 61 | \( 1 - 4.74T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 6.15T + 71T^{2} \) |
| 73 | \( 1 - 6.91T + 73T^{2} \) |
| 79 | \( 1 + 1.03T + 79T^{2} \) |
| 83 | \( 1 + 4.92T + 83T^{2} \) |
| 89 | \( 1 + 9.50T + 89T^{2} \) |
| 97 | \( 1 + 5.72T + 97T^{2} \) |
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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
−8.292938003243002534366255300632, −7.57171593667456686522658447229, −7.05987793377442657515500447503, −6.25989175406816322667072262662, −4.52665031347056741778054746747, −4.33122721289493016826850630802, −3.42673977407963361111844396418, −2.53136713290224654027666428283, −1.82406144422319686642546887955, −0.842748641692983015070243561365,
0.842748641692983015070243561365, 1.82406144422319686642546887955, 2.53136713290224654027666428283, 3.42673977407963361111844396418, 4.33122721289493016826850630802, 4.52665031347056741778054746747, 6.25989175406816322667072262662, 7.05987793377442657515500447503, 7.57171593667456686522658447229, 8.292938003243002534366255300632
