L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 11-s − 0.694·13-s + 14-s + 16-s − 2.69·17-s + 5.51·19-s − 20-s − 22-s − 7.43·23-s + 25-s − 0.694·26-s + 28-s − 0.610·29-s − 7.51·31-s + 32-s − 2.69·34-s − 35-s − 1.30·37-s + 5.51·38-s − 40-s − 9.51·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.192·13-s + 0.267·14-s + 0.250·16-s − 0.653·17-s + 1.26·19-s − 0.223·20-s − 0.213·22-s − 1.55·23-s + 0.200·25-s − 0.136·26-s + 0.188·28-s − 0.113·29-s − 1.35·31-s + 0.176·32-s − 0.462·34-s − 0.169·35-s − 0.214·37-s + 0.895·38-s − 0.158·40-s − 1.48·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 0.694T + 13T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 - 5.51T + 19T^{2} \) |
| 23 | \( 1 + 7.43T + 23T^{2} \) |
| 29 | \( 1 + 0.610T + 29T^{2} \) |
| 31 | \( 1 + 7.51T + 31T^{2} \) |
| 37 | \( 1 + 1.30T + 37T^{2} \) |
| 41 | \( 1 + 9.51T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 4.12T + 53T^{2} \) |
| 59 | \( 1 - 3.51T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 8.95T + 67T^{2} \) |
| 71 | \( 1 + 4.77T + 71T^{2} \) |
| 73 | \( 1 - 5.51T + 73T^{2} \) |
| 79 | \( 1 + 2.77T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 + 8.04T + 89T^{2} \) |
| 97 | \( 1 + 6.90T + 97T^{2} \) |
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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
−7.50880468771238657421687856846, −6.98268754488780354803736126675, −6.06286853094046987276780595104, −5.39588724426289211128259066865, −4.76602735867134274067711522181, −3.96781672673926220369698382531, −3.33974922365000319027142782428, −2.36726894870334040937903672514, −1.51132586853707474672483992571, 0,
1.51132586853707474672483992571, 2.36726894870334040937903672514, 3.33974922365000319027142782428, 3.96781672673926220369698382531, 4.76602735867134274067711522181, 5.39588724426289211128259066865, 6.06286853094046987276780595104, 6.98268754488780354803736126675, 7.50880468771238657421687856846