| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s − 2·11-s + 6·13-s + 15-s − 2·17-s − 19-s − 4·21-s − 6·23-s + 25-s − 27-s + 8·29-s + 8·31-s + 2·33-s − 4·35-s + 10·37-s − 6·39-s − 4·41-s − 4·43-s − 45-s − 6·47-s + 9·49-s + 2·51-s + 12·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.603·11-s + 1.66·13-s + 0.258·15-s − 0.485·17-s − 0.229·19-s − 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.48·29-s + 1.43·31-s + 0.348·33-s − 0.676·35-s + 1.64·37-s − 0.960·39-s − 0.624·41-s − 0.609·43-s − 0.149·45-s − 0.875·47-s + 9/7·49-s + 0.280·51-s + 1.64·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.825826327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.825826327\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.356563890293957794000139615142, −7.81438172298263957218751721591, −6.78959424761384834392599078645, −6.11443122339681034548914185781, −5.38303981621335948486439914863, −4.45597351494920758766791387660, −4.16903853251358762455100946022, −2.85853904604941301015742123630, −1.73980526447016001573246886507, −0.821477508523110054438478993132,
0.821477508523110054438478993132, 1.73980526447016001573246886507, 2.85853904604941301015742123630, 4.16903853251358762455100946022, 4.45597351494920758766791387660, 5.38303981621335948486439914863, 6.11443122339681034548914185781, 6.78959424761384834392599078645, 7.81438172298263957218751721591, 8.356563890293957794000139615142
