L(s) = 1 | + 7-s − 5·11-s + 13-s − 17-s − 6·23-s − 5·25-s + 10·29-s + 8·31-s + 9·37-s + 5·41-s − 3·43-s − 6·49-s − 4·53-s − 59-s + 6·61-s − 4·67-s + 71-s − 5·77-s + 3·79-s + 7·83-s − 16·89-s + 91-s − 12·97-s − 17·101-s − 4·103-s − 6·107-s − 10·109-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 1.50·11-s + 0.277·13-s − 0.242·17-s − 1.25·23-s − 25-s + 1.85·29-s + 1.43·31-s + 1.47·37-s + 0.780·41-s − 0.457·43-s − 6/7·49-s − 0.549·53-s − 0.130·59-s + 0.768·61-s − 0.488·67-s + 0.118·71-s − 0.569·77-s + 0.337·79-s + 0.768·83-s − 1.69·89-s + 0.104·91-s − 1.21·97-s − 1.69·101-s − 0.394·103-s − 0.580·107-s − 0.957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.73522639727208755225701539769, −6.63589372893505725152644746115, −6.09959394270070766803876469956, −5.34286894871960981597114424482, −4.61626070106655095086455498447, −4.03175056215354280787537798114, −2.84226327974215419521809610393, −2.39761796645296071179736093812, −1.23165076775551073116240947890, 0,
1.23165076775551073116240947890, 2.39761796645296071179736093812, 2.84226327974215419521809610393, 4.03175056215354280787537798114, 4.61626070106655095086455498447, 5.34286894871960981597114424482, 6.09959394270070766803876469956, 6.63589372893505725152644746115, 7.73522639727208755225701539769