L(s) = 1 | + 2·5-s − 4·13-s − 8·17-s − 25-s − 10·29-s − 12·37-s + 8·41-s − 7·49-s + 14·53-s − 12·61-s − 8·65-s + 6·73-s − 16·85-s + 16·89-s + 18·97-s − 2·101-s − 20·109-s + 16·113-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.10·13-s − 1.94·17-s − 1/5·25-s − 1.85·29-s − 1.97·37-s + 1.24·41-s − 49-s + 1.92·53-s − 1.53·61-s − 0.992·65-s + 0.702·73-s − 1.73·85-s + 1.69·89-s + 1.82·97-s − 0.199·101-s − 1.91·109-s + 1.50·113-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 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.955643677804340639523778996300, −7.74398608326178337806970095846, −7.04532327351522358604131570139, −6.28655854302767590234892421883, −5.45095421797933849120346272823, −4.71660698363879354831850755374, −3.73220900163596549184218440331, −2.41954791177205934755391933624, −1.86735723343660348363185523771, 0,
1.86735723343660348363185523771, 2.41954791177205934755391933624, 3.73220900163596549184218440331, 4.71660698363879354831850755374, 5.45095421797933849120346272823, 6.28655854302767590234892421883, 7.04532327351522358604131570139, 7.74398608326178337806970095846, 8.955643677804340639523778996300