L(s) = 1 | − 3-s − 5-s − 4.27·7-s + 9-s + 2.27·11-s + 13-s + 15-s − 6.27·17-s + 2·19-s + 4.27·21-s + 0.274·23-s + 25-s − 27-s + 4·31-s − 2.27·33-s + 4.27·35-s + 8.27·37-s − 39-s + 8.27·41-s − 8.54·43-s − 45-s + 4·47-s + 11.2·49-s + 6.27·51-s + 8.27·53-s − 2.27·55-s − 2·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.61·7-s + 0.333·9-s + 0.685·11-s + 0.277·13-s + 0.258·15-s − 1.52·17-s + 0.458·19-s + 0.932·21-s + 0.0573·23-s + 0.200·25-s − 0.192·27-s + 0.718·31-s − 0.396·33-s + 0.722·35-s + 1.36·37-s − 0.160·39-s + 1.29·41-s − 1.30·43-s − 0.149·45-s + 0.583·47-s + 1.61·49-s + 0.878·51-s + 1.13·53-s − 0.306·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4.27T + 7T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 17 | \( 1 + 6.27T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 0.274T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 8.27T + 37T^{2} \) |
| 41 | \( 1 - 8.27T + 41T^{2} \) |
| 43 | \( 1 + 8.54T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 8.27T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 1.72T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 8.27T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.38210752037629848385596224221, −6.96926448153123331453763067572, −6.18306595372787391139059052844, −5.90715877856261936646494463490, −4.62005632564933414941523493879, −4.10774909820889921455648827476, −3.27165737006581995869423310927, −2.44832484155010830843472060168, −1.02402123390177331720026377513, 0,
1.02402123390177331720026377513, 2.44832484155010830843472060168, 3.27165737006581995869423310927, 4.10774909820889921455648827476, 4.62005632564933414941523493879, 5.90715877856261936646494463490, 6.18306595372787391139059052844, 6.96926448153123331453763067572, 7.38210752037629848385596224221