| L(s) = 1 | − 0.330·2-s + 2.69·3-s − 1.89·4-s − 0.890·6-s − 3.35·7-s + 1.28·8-s + 4.24·9-s − 3.24·11-s − 5.08·12-s + 1.10·14-s + 3.35·16-s + 1.94·17-s − 1.40·18-s − 1.24·19-s − 9.02·21-s + 1.07·22-s − 2.69·23-s + 3.46·24-s + 3.35·27-s + 6.33·28-s + 3·29-s + 3.78·31-s − 3.68·32-s − 8.73·33-s − 0.644·34-s − 8.02·36-s − 1.94·37-s + ⋯ |
| L(s) = 1 | − 0.233·2-s + 1.55·3-s − 0.945·4-s − 0.363·6-s − 1.26·7-s + 0.455·8-s + 1.41·9-s − 0.978·11-s − 1.46·12-s + 0.296·14-s + 0.838·16-s + 0.472·17-s − 0.331·18-s − 0.285·19-s − 1.96·21-s + 0.228·22-s − 0.561·23-s + 0.707·24-s + 0.645·27-s + 1.19·28-s + 0.557·29-s + 0.679·31-s − 0.651·32-s − 1.52·33-s − 0.110·34-s − 1.33·36-s − 0.320·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.741343184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.741343184\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.330T + 2T^{2} \) |
| 3 | \( 1 - 2.69T + 3T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 17 | \( 1 - 1.94T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 + 2.69T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 3.78T + 31T^{2} \) |
| 37 | \( 1 + 1.94T + 37T^{2} \) |
| 41 | \( 1 - 2.78T + 41T^{2} \) |
| 43 | \( 1 - 8.73T + 43T^{2} \) |
| 47 | \( 1 - 6.86T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 + 7.49T + 61T^{2} \) |
| 67 | \( 1 - 4.01T + 67T^{2} \) |
| 71 | \( 1 - 5.24T + 71T^{2} \) |
| 73 | \( 1 - 5.46T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 + 8.61T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 5.26T + 97T^{2} \) |
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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
−8.466232949279178742051184228836, −7.85859677688914886892558892150, −7.31338766851552513300334684505, −6.24851203289653580989091173884, −5.38953681112973522407084075785, −4.34499663368212243301087585442, −3.68560722674909413384892988328, −2.96492018464278557592206911827, −2.24040830751892930348700397859, −0.70431289429278213125420356361,
0.70431289429278213125420356361, 2.24040830751892930348700397859, 2.96492018464278557592206911827, 3.68560722674909413384892988328, 4.34499663368212243301087585442, 5.38953681112973522407084075785, 6.24851203289653580989091173884, 7.31338766851552513300334684505, 7.85859677688914886892558892150, 8.466232949279178742051184228836
