L(s) = 1 | − 3-s − 1.04·7-s + 9-s + 6.28·11-s + 1.00·13-s + 4.69·17-s − 5.97·19-s + 1.04·21-s − 8.05·23-s − 27-s + 6.91·29-s − 9.52·31-s − 6.28·33-s − 7.69·37-s − 1.00·39-s + 1.56·41-s − 9.94·43-s + 4.84·47-s − 5.90·49-s − 4.69·51-s − 2.82·53-s + 5.97·57-s + 4.08·59-s + 3.16·61-s − 1.04·63-s − 3.17·67-s + 8.05·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.395·7-s + 0.333·9-s + 1.89·11-s + 0.279·13-s + 1.13·17-s − 1.37·19-s + 0.228·21-s − 1.67·23-s − 0.192·27-s + 1.28·29-s − 1.71·31-s − 1.09·33-s − 1.26·37-s − 0.161·39-s + 0.245·41-s − 1.51·43-s + 0.706·47-s − 0.843·49-s − 0.656·51-s − 0.388·53-s + 0.791·57-s + 0.531·59-s + 0.404·61-s − 0.131·63-s − 0.387·67-s + 0.969·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.04T + 7T^{2} \) |
| 11 | \( 1 - 6.28T + 11T^{2} \) |
| 13 | \( 1 - 1.00T + 13T^{2} \) |
| 17 | \( 1 - 4.69T + 17T^{2} \) |
| 19 | \( 1 + 5.97T + 19T^{2} \) |
| 23 | \( 1 + 8.05T + 23T^{2} \) |
| 29 | \( 1 - 6.91T + 29T^{2} \) |
| 31 | \( 1 + 9.52T + 31T^{2} \) |
| 37 | \( 1 + 7.69T + 37T^{2} \) |
| 41 | \( 1 - 1.56T + 41T^{2} \) |
| 43 | \( 1 + 9.94T + 43T^{2} \) |
| 47 | \( 1 - 4.84T + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 - 4.08T + 59T^{2} \) |
| 61 | \( 1 - 3.16T + 61T^{2} \) |
| 67 | \( 1 + 3.17T + 67T^{2} \) |
| 71 | \( 1 - 6.50T + 71T^{2} \) |
| 73 | \( 1 - 0.367T + 73T^{2} \) |
| 79 | \( 1 + 3.32T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 2.50T + 89T^{2} \) |
| 97 | \( 1 + 0.182T + 97T^{2} \) |
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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
−7.36315841644195330877396795762, −6.67635070333293879820086330033, −6.22291988920368948775848622613, −5.62730207018729749050286941126, −4.63617545667775201113918813622, −3.86166558830179376252307878501, −3.44507361895546288794707806188, −2.00840960420037672480738000723, −1.28654955273594499198136991952, 0,
1.28654955273594499198136991952, 2.00840960420037672480738000723, 3.44507361895546288794707806188, 3.86166558830179376252307878501, 4.63617545667775201113918813622, 5.62730207018729749050286941126, 6.22291988920368948775848622613, 6.67635070333293879820086330033, 7.36315841644195330877396795762