L(s) = 1 | − 2.97·3-s + 0.840·5-s − 4.22·7-s + 5.85·9-s + 3.71·11-s − 1.07·13-s − 2.50·15-s + 1.37·17-s − 19-s + 12.5·21-s − 4.51·23-s − 4.29·25-s − 8.48·27-s + 0.356·29-s − 4.24·31-s − 11.0·33-s − 3.54·35-s + 4.55·37-s + 3.18·39-s + 10.9·41-s + 5.17·43-s + 4.91·45-s − 4.68·47-s + 10.8·49-s − 4.09·51-s + 53-s + 3.12·55-s + ⋯ |
L(s) = 1 | − 1.71·3-s + 0.375·5-s − 1.59·7-s + 1.95·9-s + 1.12·11-s − 0.296·13-s − 0.645·15-s + 0.333·17-s − 0.229·19-s + 2.74·21-s − 0.941·23-s − 0.858·25-s − 1.63·27-s + 0.0662·29-s − 0.763·31-s − 1.92·33-s − 0.599·35-s + 0.749·37-s + 0.509·39-s + 1.71·41-s + 0.789·43-s + 0.733·45-s − 0.683·47-s + 1.54·49-s − 0.572·51-s + 0.137·53-s + 0.421·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
good | 3 | \( 1 + 2.97T + 3T^{2} \) |
| 5 | \( 1 - 0.840T + 5T^{2} \) |
| 7 | \( 1 + 4.22T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 + 1.07T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 23 | \( 1 + 4.51T + 23T^{2} \) |
| 29 | \( 1 - 0.356T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 - 4.55T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 5.17T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 59 | \( 1 - 7.35T + 59T^{2} \) |
| 61 | \( 1 - 6.24T + 61T^{2} \) |
| 67 | \( 1 - 3.54T + 67T^{2} \) |
| 71 | \( 1 + 5.04T + 71T^{2} \) |
| 73 | \( 1 - 5.95T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 3.23T + 83T^{2} \) |
| 89 | \( 1 + 0.365T + 89T^{2} \) |
| 97 | \( 1 + 4.23T + 97T^{2} \) |
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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
−7.84992365917008550012648113800, −6.95875383111751815314854870296, −6.42918736166815419783196496637, −5.95169830539232132833040998741, −5.41222288148006338930809174909, −4.22674616773108368194116278376, −3.72521336419596382515975523680, −2.35121928740468420458617227832, −1.05065915857955406434796422669, 0,
1.05065915857955406434796422669, 2.35121928740468420458617227832, 3.72521336419596382515975523680, 4.22674616773108368194116278376, 5.41222288148006338930809174909, 5.95169830539232132833040998741, 6.42918736166815419783196496637, 6.95875383111751815314854870296, 7.84992365917008550012648113800