| L(s) = 1 | + 1.95·3-s − 3.45·5-s − 1.56·7-s + 0.828·9-s + 0.456·11-s + 4.08·13-s − 6.76·15-s + 6.28·17-s − 6.06·19-s − 3.05·21-s − 1.73·23-s + 6.93·25-s − 4.24·27-s + 5.41·29-s − 6.52·31-s + 0.893·33-s + 5.39·35-s − 9.81·37-s + 8.00·39-s + 10.9·41-s + 12.6·43-s − 2.86·45-s − 8.17·47-s − 4.56·49-s + 12.3·51-s + 8.57·53-s − 1.57·55-s + ⋯ |
| L(s) = 1 | + 1.12·3-s − 1.54·5-s − 0.589·7-s + 0.276·9-s + 0.137·11-s + 1.13·13-s − 1.74·15-s + 1.52·17-s − 1.39·19-s − 0.666·21-s − 0.360·23-s + 1.38·25-s − 0.817·27-s + 1.00·29-s − 1.17·31-s + 0.155·33-s + 0.911·35-s − 1.61·37-s + 1.28·39-s + 1.70·41-s + 1.92·43-s − 0.426·45-s − 1.19·47-s − 0.652·49-s + 1.72·51-s + 1.17·53-s − 0.212·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 1.95T + 3T^{2} \) |
| 5 | \( 1 + 3.45T + 5T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 - 0.456T + 11T^{2} \) |
| 13 | \( 1 - 4.08T + 13T^{2} \) |
| 17 | \( 1 - 6.28T + 17T^{2} \) |
| 19 | \( 1 + 6.06T + 19T^{2} \) |
| 23 | \( 1 + 1.73T + 23T^{2} \) |
| 29 | \( 1 - 5.41T + 29T^{2} \) |
| 31 | \( 1 + 6.52T + 31T^{2} \) |
| 37 | \( 1 + 9.81T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 + 8.17T + 47T^{2} \) |
| 53 | \( 1 - 8.57T + 53T^{2} \) |
| 59 | \( 1 - 5.54T + 59T^{2} \) |
| 61 | \( 1 + 0.395T + 61T^{2} \) |
| 67 | \( 1 - 1.92T + 67T^{2} \) |
| 71 | \( 1 - 1.21T + 71T^{2} \) |
| 73 | \( 1 - 5.21T + 73T^{2} \) |
| 79 | \( 1 + 7.52T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + 9.84T + 89T^{2} \) |
| 97 | \( 1 + 1.24T + 97T^{2} \) |
| 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
−7.65904016304225860974857522780, −7.03780431242578643942039320764, −6.19259068792110955024516527958, −5.42853000823676668106144841370, −4.13103870568575809263420094168, −3.86475307189785632556193706310, −3.25946774461457947064390828259, −2.50605806979641504389341012665, −1.24538119170006638485584720299, 0,
1.24538119170006638485584720299, 2.50605806979641504389341012665, 3.25946774461457947064390828259, 3.86475307189785632556193706310, 4.13103870568575809263420094168, 5.42853000823676668106144841370, 6.19259068792110955024516527958, 7.03780431242578643942039320764, 7.65904016304225860974857522780
