L(s) = 1 | − 2-s − 3-s + 4-s + 3.93·5-s + 6-s − 7-s − 8-s + 9-s − 3.93·10-s + 4.81·11-s − 12-s + 14-s − 3.93·15-s + 16-s − 3.61·17-s − 18-s − 7.71·19-s + 3.93·20-s + 21-s − 4.81·22-s − 3.86·23-s + 24-s + 10.4·25-s − 27-s − 28-s − 4.45·29-s + 3.93·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.75·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.24·10-s + 1.45·11-s − 0.288·12-s + 0.267·14-s − 1.01·15-s + 0.250·16-s − 0.876·17-s − 0.235·18-s − 1.76·19-s + 0.879·20-s + 0.218·21-s − 1.02·22-s − 0.805·23-s + 0.204·24-s + 2.09·25-s − 0.192·27-s − 0.188·28-s − 0.826·29-s + 0.718·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 3.93T + 5T^{2} \) |
| 11 | \( 1 - 4.81T + 11T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 + 7.71T + 19T^{2} \) |
| 23 | \( 1 + 3.86T + 23T^{2} \) |
| 29 | \( 1 + 4.45T + 29T^{2} \) |
| 31 | \( 1 - 1.01T + 31T^{2} \) |
| 37 | \( 1 - 7.30T + 37T^{2} \) |
| 41 | \( 1 - 7.28T + 41T^{2} \) |
| 43 | \( 1 + 4.73T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 3.70T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 3.12T + 67T^{2} \) |
| 71 | \( 1 + 8.73T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 + 0.731T + 79T^{2} \) |
| 83 | \( 1 + 0.380T + 83T^{2} \) |
| 89 | \( 1 - 0.469T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.48969861925673915806585517733, −6.53245076603155795573389820123, −6.23082926290354785123012878242, −6.00622011767256482415884695307, −4.77446372548933170046820512537, −4.09366979716204584477657808103, −2.83848979248380154864606527743, −1.90085968206951949521185575291, −1.45717367898828177798531691910, 0,
1.45717367898828177798531691910, 1.90085968206951949521185575291, 2.83848979248380154864606527743, 4.09366979716204584477657808103, 4.77446372548933170046820512537, 6.00622011767256482415884695307, 6.23082926290354785123012878242, 6.53245076603155795573389820123, 7.48969861925673915806585517733