| L(s) = 1 | − 2.68·3-s + 4.09·5-s − 1.54·7-s + 4.19·9-s − 2.79·11-s − 4.67·13-s − 10.9·15-s + 2.25·17-s + 4.13·21-s − 0.190·23-s + 11.7·25-s − 3.21·27-s − 2.54·29-s + 7.90·31-s + 7.49·33-s − 6.30·35-s − 9.21·37-s + 12.5·39-s + 0.0643·41-s + 5.52·43-s + 17.1·45-s + 5.89·47-s − 4.62·49-s − 6.03·51-s + 11.1·53-s − 11.4·55-s − 7.15·59-s + ⋯ |
| L(s) = 1 | − 1.54·3-s + 1.83·5-s − 0.582·7-s + 1.39·9-s − 0.842·11-s − 1.29·13-s − 2.83·15-s + 0.545·17-s + 0.902·21-s − 0.0396·23-s + 2.35·25-s − 0.618·27-s − 0.471·29-s + 1.41·31-s + 1.30·33-s − 1.06·35-s − 1.51·37-s + 2.00·39-s + 0.0100·41-s + 0.841·43-s + 2.56·45-s + 0.860·47-s − 0.660·49-s − 0.845·51-s + 1.52·53-s − 1.54·55-s − 0.930·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.68T + 3T^{2} \) |
| 5 | \( 1 - 4.09T + 5T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 + 4.67T + 13T^{2} \) |
| 17 | \( 1 - 2.25T + 17T^{2} \) |
| 23 | \( 1 + 0.190T + 23T^{2} \) |
| 29 | \( 1 + 2.54T + 29T^{2} \) |
| 31 | \( 1 - 7.90T + 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 - 0.0643T + 41T^{2} \) |
| 43 | \( 1 - 5.52T + 43T^{2} \) |
| 47 | \( 1 - 5.89T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 7.15T + 59T^{2} \) |
| 61 | \( 1 + 1.06T + 61T^{2} \) |
| 67 | \( 1 - 0.397T + 67T^{2} \) |
| 71 | \( 1 + 1.74T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 6.67T + 83T^{2} \) |
| 89 | \( 1 + 0.659T + 89T^{2} \) |
| 97 | \( 1 - 5.40T + 97T^{2} \) |
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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
−7.36559813900694193102850586322, −6.89797313766465503514873871661, −6.05836917383628272028632827768, −5.69525375608732161758245522229, −5.14399632984543839316517998940, −4.51356236461310574776213464975, −2.98590761583601980739595614161, −2.27478649665374026848992550507, −1.20094037260526200543413804700, 0,
1.20094037260526200543413804700, 2.27478649665374026848992550507, 2.98590761583601980739595614161, 4.51356236461310574776213464975, 5.14399632984543839316517998940, 5.69525375608732161758245522229, 6.05836917383628272028632827768, 6.89797313766465503514873871661, 7.36559813900694193102850586322
