| L(s) = 1 | + 1.10·3-s − 5-s − 2.46·7-s − 1.77·9-s − 1.71·11-s + 6.49·13-s − 1.10·15-s + 3.32·17-s − 0.734·19-s − 2.72·21-s + 2.08·23-s + 25-s − 5.28·27-s − 4.21·29-s − 7.46·31-s − 1.89·33-s + 2.46·35-s + 37-s + 7.17·39-s + 1.71·41-s − 1.81·43-s + 1.77·45-s + 0.882·47-s − 0.940·49-s + 3.68·51-s − 7.03·53-s + 1.71·55-s + ⋯ |
| L(s) = 1 | + 0.638·3-s − 0.447·5-s − 0.930·7-s − 0.592·9-s − 0.517·11-s + 1.80·13-s − 0.285·15-s + 0.807·17-s − 0.168·19-s − 0.593·21-s + 0.434·23-s + 0.200·25-s − 1.01·27-s − 0.782·29-s − 1.34·31-s − 0.330·33-s + 0.416·35-s + 0.164·37-s + 1.14·39-s + 0.267·41-s − 0.277·43-s + 0.264·45-s + 0.128·47-s − 0.134·49-s + 0.515·51-s − 0.965·53-s + 0.231·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 1.10T + 3T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 + 1.71T + 11T^{2} \) |
| 13 | \( 1 - 6.49T + 13T^{2} \) |
| 17 | \( 1 - 3.32T + 17T^{2} \) |
| 19 | \( 1 + 0.734T + 19T^{2} \) |
| 23 | \( 1 - 2.08T + 23T^{2} \) |
| 29 | \( 1 + 4.21T + 29T^{2} \) |
| 31 | \( 1 + 7.46T + 31T^{2} \) |
| 41 | \( 1 - 1.71T + 41T^{2} \) |
| 43 | \( 1 + 1.81T + 43T^{2} \) |
| 47 | \( 1 - 0.882T + 47T^{2} \) |
| 53 | \( 1 + 7.03T + 53T^{2} \) |
| 59 | \( 1 + 0.387T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 + 8.23T + 79T^{2} \) |
| 83 | \( 1 - 4.80T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−8.318229365740873942860938157247, −7.82081141380022946289323497595, −6.92206813233682959022221625955, −6.00083453941299709470696651656, −5.50161858650883499758728825204, −4.14573447902541874753868418282, −3.36780371711373686603880504702, −2.95541622176358851952492277817, −1.54035207492217391474610803279, 0,
1.54035207492217391474610803279, 2.95541622176358851952492277817, 3.36780371711373686603880504702, 4.14573447902541874753868418282, 5.50161858650883499758728825204, 6.00083453941299709470696651656, 6.92206813233682959022221625955, 7.82081141380022946289323497595, 8.318229365740873942860938157247
