L(s) = 1 | − 3-s − 4.02·5-s − 4.61·7-s + 9-s + 1.53·11-s + 5.57·13-s + 4.02·15-s − 1.29·17-s + 4.35·19-s + 4.61·21-s − 4·23-s + 11.2·25-s − 27-s + 1.86·29-s + 2.77·31-s − 1.53·33-s + 18.5·35-s − 3.97·37-s − 5.57·39-s − 3.03·41-s + 3.03·43-s − 4.02·45-s − 9.65·47-s + 14.2·49-s + 1.29·51-s + 8.58·53-s − 6.16·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.80·5-s − 1.74·7-s + 0.333·9-s + 0.461·11-s + 1.54·13-s + 1.03·15-s − 0.314·17-s + 1.00·19-s + 1.00·21-s − 0.834·23-s + 2.24·25-s − 0.192·27-s + 0.345·29-s + 0.498·31-s − 0.266·33-s + 3.14·35-s − 0.653·37-s − 0.893·39-s − 0.473·41-s + 0.462·43-s − 0.600·45-s − 1.40·47-s + 2.04·49-s + 0.181·51-s + 1.17·53-s − 0.831·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 5 | \( 1 + 4.02T + 5T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 - 5.57T + 13T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 19 | \( 1 - 4.35T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 1.86T + 29T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 + 3.97T + 37T^{2} \) |
| 41 | \( 1 + 3.03T + 41T^{2} \) |
| 43 | \( 1 - 3.03T + 43T^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 - 8.58T + 53T^{2} \) |
| 59 | \( 1 - 5.73T + 59T^{2} \) |
| 61 | \( 1 - 7.64T + 61T^{2} \) |
| 67 | \( 1 + 8.79T + 67T^{2} \) |
| 71 | \( 1 - 6.88T + 71T^{2} \) |
| 73 | \( 1 + 3.50T + 73T^{2} \) |
| 79 | \( 1 + 8.18T + 79T^{2} \) |
| 83 | \( 1 - 4.12T + 83T^{2} \) |
| 89 | \( 1 - 7.98T + 89T^{2} \) |
| 97 | \( 1 + 1.40T + 97T^{2} \) |
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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
−8.357271276631848232791601627293, −7.48533755968911709383519947980, −6.69808100888693796744818659811, −6.32744752741606613608783378347, −5.29476200871291047121097128844, −4.04523083541068841246733150332, −3.74443548375038028090759043329, −2.98135627378011384545251074075, −1.02935264448947798211128901563, 0,
1.02935264448947798211128901563, 2.98135627378011384545251074075, 3.74443548375038028090759043329, 4.04523083541068841246733150332, 5.29476200871291047121097128844, 6.32744752741606613608783378347, 6.69808100888693796744818659811, 7.48533755968911709383519947980, 8.357271276631848232791601627293