| L(s) = 1 | + 0.239·2-s − 1.94·4-s + 2.59·5-s − 0.942·8-s + 0.619·10-s + 4.18·11-s − 3.68·13-s + 3.66·16-s − 1.71·17-s + 7.15·19-s − 5.03·20-s + 0.999·22-s − 5.12·23-s + 1.71·25-s − 0.880·26-s − 2.12·29-s + 6.53·31-s + 2.76·32-s − 0.409·34-s + 1.66·37-s + 1.71·38-s − 2.44·40-s + 10.2·41-s − 1.66·43-s − 8.12·44-s − 1.22·46-s − 9.33·47-s + ⋯ |
| L(s) = 1 | + 0.169·2-s − 0.971·4-s + 1.15·5-s − 0.333·8-s + 0.195·10-s + 1.26·11-s − 1.02·13-s + 0.915·16-s − 0.414·17-s + 1.64·19-s − 1.12·20-s + 0.213·22-s − 1.06·23-s + 0.343·25-s − 0.172·26-s − 0.394·29-s + 1.17·31-s + 0.488·32-s − 0.0701·34-s + 0.272·37-s + 0.277·38-s − 0.386·40-s + 1.59·41-s − 0.253·43-s − 1.22·44-s − 0.180·46-s − 1.36·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.116974922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.116974922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.239T + 2T^{2} \) |
| 5 | \( 1 - 2.59T + 5T^{2} \) |
| 11 | \( 1 - 4.18T + 11T^{2} \) |
| 13 | \( 1 + 3.68T + 13T^{2} \) |
| 17 | \( 1 + 1.71T + 17T^{2} \) |
| 19 | \( 1 - 7.15T + 19T^{2} \) |
| 23 | \( 1 + 5.12T + 23T^{2} \) |
| 29 | \( 1 + 2.12T + 29T^{2} \) |
| 31 | \( 1 - 6.53T + 31T^{2} \) |
| 37 | \( 1 - 1.66T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 1.66T + 43T^{2} \) |
| 47 | \( 1 + 9.33T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 6.06T + 59T^{2} \) |
| 61 | \( 1 - 7.98T + 61T^{2} \) |
| 67 | \( 1 - 8.26T + 67T^{2} \) |
| 71 | \( 1 - 6.23T + 71T^{2} \) |
| 73 | \( 1 + 7.15T + 73T^{2} \) |
| 79 | \( 1 + 9.82T + 79T^{2} \) |
| 83 | \( 1 + 6.89T + 83T^{2} \) |
| 89 | \( 1 - 5.03T + 89T^{2} \) |
| 97 | \( 1 + 3.06T + 97T^{2} \) |
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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
−8.616339608872218807001591977282, −7.76718391780384055417301478172, −6.90608691879981599547920659182, −6.03819562244166464195224065210, −5.52877690683411047793576601994, −4.69317703753474246670580770909, −3.99676714357992567559688925848, −2.98976702205115896092848855908, −1.93658296258068489834258856543, −0.849636426600035289535070129240,
0.849636426600035289535070129240, 1.93658296258068489834258856543, 2.98976702205115896092848855908, 3.99676714357992567559688925848, 4.69317703753474246670580770909, 5.52877690683411047793576601994, 6.03819562244166464195224065210, 6.90608691879981599547920659182, 7.76718391780384055417301478172, 8.616339608872218807001591977282
