| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 6·11-s + 12-s + 13-s − 15-s + 16-s + 18-s − 7·19-s − 20-s − 6·22-s + 24-s + 25-s + 26-s + 27-s + 6·29-s − 30-s − 8·31-s + 32-s − 6·33-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.80·11-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 1.60·19-s − 0.223·20-s − 1.27·22-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 1.11·29-s − 0.182·30-s − 1.43·31-s + 0.176·32-s − 1.04·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.23512336332078, −12.80960710303265, −12.45954255260704, −12.16270430480726, −11.29052411105310, −10.86919014962025, −10.58820099663566, −10.20248418008071, −9.520434438088591, −8.760245521130473, −8.560594929687873, −8.017784307021825, −7.399476897835632, −7.271389082141524, −6.493797216923374, −5.929177766153494, −5.499878358408991, −4.836109630373675, −4.440132969797672, −3.921481988541060, −3.310153320199309, −2.805421914124977, −2.263275254231089, −1.844873268423982, −0.7907488274319100, 0,
0.7907488274319100, 1.844873268423982, 2.263275254231089, 2.805421914124977, 3.310153320199309, 3.921481988541060, 4.440132969797672, 4.836109630373675, 5.499878358408991, 5.929177766153494, 6.493797216923374, 7.271389082141524, 7.399476897835632, 8.017784307021825, 8.560594929687873, 8.760245521130473, 9.520434438088591, 10.20248418008071, 10.58820099663566, 10.86919014962025, 11.29052411105310, 12.16270430480726, 12.45954255260704, 12.80960710303265, 13.23512336332078
