| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 4.47·13-s + 15-s − 4.47·17-s − 21-s + 6.47·23-s + 25-s − 27-s − 8.47·29-s − 6.47·31-s − 35-s − 8.47·37-s − 4.47·39-s + 10.9·41-s − 10.4·43-s − 45-s − 2.47·47-s + 49-s + 4.47·51-s − 2·53-s + 4·59-s − 12.4·61-s + 63-s − 4.47·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s + 1.24·13-s + 0.258·15-s − 1.08·17-s − 0.218·21-s + 1.34·23-s + 0.200·25-s − 0.192·27-s − 1.57·29-s − 1.16·31-s − 0.169·35-s − 1.39·37-s − 0.716·39-s + 1.70·41-s − 1.59·43-s − 0.149·45-s − 0.360·47-s + 0.142·49-s + 0.626·51-s − 0.274·53-s + 0.520·59-s − 1.59·61-s + 0.125·63-s − 0.554·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 2.47T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 - 8.47T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 6.94T + 89T^{2} \) |
| 97 | \( 1 + 4.47T + 97T^{2} \) |
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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
−7.58400705768765675929275648209, −6.86185048860045792638341218276, −6.32014825213729218481076959664, −5.38938401713916874867455528795, −4.90500382662263215337757360449, −3.92537317304507086089953496209, −3.43444876228649193080650819358, −2.12506417643249451370148878656, −1.23320469953243744914173468680, 0,
1.23320469953243744914173468680, 2.12506417643249451370148878656, 3.43444876228649193080650819358, 3.92537317304507086089953496209, 4.90500382662263215337757360449, 5.38938401713916874867455528795, 6.32014825213729218481076959664, 6.86185048860045792638341218276, 7.58400705768765675929275648209
