| L(s) = 1 | − 0.445·2-s − 1.80·4-s − 5-s − 1.44·7-s + 1.69·8-s + 0.445·10-s − 2.60·11-s + 0.643·14-s + 2.85·16-s + 4.89·17-s − 4.80·19-s + 1.80·20-s + 1.15·22-s + 4.75·23-s + 25-s + 2.60·28-s + 0.594·29-s − 5.31·31-s − 4.65·32-s − 2.17·34-s + 1.44·35-s + 1.08·37-s + 2.13·38-s − 1.69·40-s − 2.55·41-s − 2.77·43-s + 4.69·44-s + ⋯ |
| L(s) = 1 | − 0.314·2-s − 0.900·4-s − 0.447·5-s − 0.546·7-s + 0.598·8-s + 0.140·10-s − 0.785·11-s + 0.171·14-s + 0.712·16-s + 1.18·17-s − 1.10·19-s + 0.402·20-s + 0.247·22-s + 0.991·23-s + 0.200·25-s + 0.492·28-s + 0.110·29-s − 0.955·31-s − 0.822·32-s − 0.373·34-s + 0.244·35-s + 0.178·37-s + 0.346·38-s − 0.267·40-s − 0.399·41-s − 0.423·43-s + 0.707·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6302936658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6302936658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.445T + 2T^{2} \) |
| 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 4.80T + 19T^{2} \) |
| 23 | \( 1 - 4.75T + 23T^{2} \) |
| 29 | \( 1 - 0.594T + 29T^{2} \) |
| 31 | \( 1 + 5.31T + 31T^{2} \) |
| 37 | \( 1 - 1.08T + 37T^{2} \) |
| 41 | \( 1 + 2.55T + 41T^{2} \) |
| 43 | \( 1 + 2.77T + 43T^{2} \) |
| 47 | \( 1 - 2.43T + 47T^{2} \) |
| 53 | \( 1 + 3.82T + 53T^{2} \) |
| 59 | \( 1 - 3.65T + 59T^{2} \) |
| 61 | \( 1 + 9.45T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 4.80T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 + 0.868T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.85687893692325844875308592222, −7.43998264596245201391465550550, −6.55080611136157150985840612505, −5.66891780971876292198479570516, −5.04877886437259546564404789988, −4.33235826277760742925931363629, −3.51384677415762352030516292712, −2.88682594765920156732307222465, −1.57514636326186391676416074436, −0.42881316374830223748502724276,
0.42881316374830223748502724276, 1.57514636326186391676416074436, 2.88682594765920156732307222465, 3.51384677415762352030516292712, 4.33235826277760742925931363629, 5.04877886437259546564404789988, 5.66891780971876292198479570516, 6.55080611136157150985840612505, 7.43998264596245201391465550550, 7.85687893692325844875308592222
