| L(s) = 1 | − 3.30·3-s + 5-s + 0.302·7-s + 7.90·9-s + 3.30·11-s + 2.30·13-s − 3.30·15-s + 7.30·17-s − 2.90·19-s − 1.00·21-s + 23-s + 25-s − 16.2·27-s − 8.60·29-s − 4.30·31-s − 10.9·33-s + 0.302·35-s − 5.21·37-s − 7.60·39-s − 4.69·41-s − 4·43-s + 7.90·45-s − 3.39·47-s − 6.90·49-s − 24.1·51-s + 7.21·53-s + 3.30·55-s + ⋯ |
| L(s) = 1 | − 1.90·3-s + 0.447·5-s + 0.114·7-s + 2.63·9-s + 0.995·11-s + 0.638·13-s − 0.852·15-s + 1.77·17-s − 0.667·19-s − 0.218·21-s + 0.208·23-s + 0.200·25-s − 3.11·27-s − 1.59·29-s − 0.772·31-s − 1.89·33-s + 0.0511·35-s − 0.856·37-s − 1.21·39-s − 0.733·41-s − 0.609·43-s + 1.17·45-s − 0.495·47-s − 0.986·49-s − 3.37·51-s + 0.990·53-s + 0.445·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 7 | \( 1 - 0.302T + 7T^{2} \) |
| 11 | \( 1 - 3.30T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 - 7.30T + 17T^{2} \) |
| 19 | \( 1 + 2.90T + 19T^{2} \) |
| 29 | \( 1 + 8.60T + 29T^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 + 5.21T + 37T^{2} \) |
| 41 | \( 1 + 4.69T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 3.39T + 47T^{2} \) |
| 53 | \( 1 - 7.21T + 53T^{2} \) |
| 59 | \( 1 + 1.39T + 59T^{2} \) |
| 61 | \( 1 + 1.30T + 61T^{2} \) |
| 67 | \( 1 + 9.21T + 67T^{2} \) |
| 71 | \( 1 + 5.30T + 71T^{2} \) |
| 73 | \( 1 - 5.39T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 2.78T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.22266349263049605087810173676, −6.73292998674861013119023952939, −6.00932208788318772113606105425, −5.56827772788063819312214295772, −5.00632052792353410505653707057, −4.05146868372649466566531579941, −3.44592046486286322546217075548, −1.68641615820823444187673095497, −1.28398725105889730217362062664, 0,
1.28398725105889730217362062664, 1.68641615820823444187673095497, 3.44592046486286322546217075548, 4.05146868372649466566531579941, 5.00632052792353410505653707057, 5.56827772788063819312214295772, 6.00932208788318772113606105425, 6.73292998674861013119023952939, 7.22266349263049605087810173676
