| L(s) = 1 | − 2.82·11-s − 6·17-s + 8.48·19-s − 5·25-s − 6·41-s − 8.48·43-s − 7·49-s − 14.1·59-s + 8.48·67-s + 2·73-s + 2.82·83-s − 18·89-s − 10·97-s + 19.7·107-s − 18·113-s + ⋯ |
| L(s) = 1 | − 0.852·11-s − 1.45·17-s + 1.94·19-s − 25-s − 0.937·41-s − 1.29·43-s − 49-s − 1.84·59-s + 1.03·67-s + 0.234·73-s + 0.310·83-s − 1.90·89-s − 1.01·97-s + 1.91·107-s − 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 8.48T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.550604707190287457449162048205, −7.86212119534293610670475468169, −7.13329030857311074864957683359, −6.30756043982600969129568248348, −5.34994131192099169066173030259, −4.74925535467413713567055616405, −3.61700739067622439442995390282, −2.73825871353149658626634291125, −1.62541783410537901570397045449, 0,
1.62541783410537901570397045449, 2.73825871353149658626634291125, 3.61700739067622439442995390282, 4.74925535467413713567055616405, 5.34994131192099169066173030259, 6.30756043982600969129568248348, 7.13329030857311074864957683359, 7.86212119534293610670475468169, 8.550604707190287457449162048205
