L(s) = 1 | + 2.77·3-s − 1.91·5-s − 1.46·7-s + 4.69·9-s − 2.45·11-s + 0.468·13-s − 5.30·15-s − 17-s + 1.59·19-s − 4.05·21-s + 2.10·23-s − 1.34·25-s + 4.71·27-s − 1.76·29-s − 2.94·31-s − 6.80·33-s + 2.79·35-s + 2.38·37-s + 1.30·39-s + 6.96·41-s + 6.83·43-s − 8.98·45-s + 3.84·47-s − 4.86·49-s − 2.77·51-s − 3.98·53-s + 4.68·55-s + ⋯ |
L(s) = 1 | + 1.60·3-s − 0.855·5-s − 0.552·7-s + 1.56·9-s − 0.739·11-s + 0.129·13-s − 1.37·15-s − 0.242·17-s + 0.366·19-s − 0.885·21-s + 0.439·23-s − 0.268·25-s + 0.907·27-s − 0.327·29-s − 0.528·31-s − 1.18·33-s + 0.472·35-s + 0.392·37-s + 0.208·39-s + 1.08·41-s + 1.04·43-s − 1.33·45-s + 0.560·47-s − 0.694·49-s − 0.388·51-s − 0.547·53-s + 0.632·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 - 2.77T + 3T^{2} \) |
| 5 | \( 1 + 1.91T + 5T^{2} \) |
| 7 | \( 1 + 1.46T + 7T^{2} \) |
| 11 | \( 1 + 2.45T + 11T^{2} \) |
| 13 | \( 1 - 0.468T + 13T^{2} \) |
| 19 | \( 1 - 1.59T + 19T^{2} \) |
| 23 | \( 1 - 2.10T + 23T^{2} \) |
| 29 | \( 1 + 1.76T + 29T^{2} \) |
| 31 | \( 1 + 2.94T + 31T^{2} \) |
| 37 | \( 1 - 2.38T + 37T^{2} \) |
| 41 | \( 1 - 6.96T + 41T^{2} \) |
| 43 | \( 1 - 6.83T + 43T^{2} \) |
| 47 | \( 1 - 3.84T + 47T^{2} \) |
| 53 | \( 1 + 3.98T + 53T^{2} \) |
| 61 | \( 1 + 2.87T + 61T^{2} \) |
| 67 | \( 1 + 9.15T + 67T^{2} \) |
| 71 | \( 1 + 16.4T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 6.65T + 79T^{2} \) |
| 83 | \( 1 + 7.60T + 83T^{2} \) |
| 89 | \( 1 - 3.95T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.58798372753205585651040276126, −7.21671964857153986085712813737, −6.21061536284085372129318779386, −5.32470372443339334077290699526, −4.25425971568649581408463121134, −3.86165852959627289284261896262, −2.95634980792359799494599824881, −2.61254204478258227348005407589, −1.44540929860504133809559183375, 0,
1.44540929860504133809559183375, 2.61254204478258227348005407589, 2.95634980792359799494599824881, 3.86165852959627289284261896262, 4.25425971568649581408463121134, 5.32470372443339334077290699526, 6.21061536284085372129318779386, 7.21671964857153986085712813737, 7.58798372753205585651040276126