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
\(L(1)\) |
\(\approx\) |
\(0.3738432391\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3738432391\) |
\(L(\frac{3}{2})\) |
|
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
−8.618404624579347235932738271882, −7.48618427835435461530158126395, −6.64131276134834124388001638790, −6.32057495322929194154195324533, −5.41913267534720401038363108007, −4.80192235225130960344163401616, −3.97996591294336267499799819353, −3.18813586794330586070947114441, −1.44183145116826615498407684104, −0.42144128874520658845038531278,
0.42144128874520658845038531278, 1.44183145116826615498407684104, 3.18813586794330586070947114441, 3.97996591294336267499799819353, 4.80192235225130960344163401616, 5.41913267534720401038363108007, 6.32057495322929194154195324533, 6.64131276134834124388001638790, 7.48618427835435461530158126395, 8.618404624579347235932738271882