L(s) = 1 | + 4·5-s − 4·13-s + 2·17-s + 11·25-s + 4·29-s + 12·37-s + 10·41-s − 7·49-s + 4·53-s + 12·61-s − 16·65-s − 6·73-s + 8·85-s − 10·89-s − 18·97-s + 20·101-s − 20·109-s + 14·113-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 1.10·13-s + 0.485·17-s + 11/5·25-s + 0.742·29-s + 1.97·37-s + 1.56·41-s − 49-s + 0.549·53-s + 1.53·61-s − 1.98·65-s − 0.702·73-s + 0.867·85-s − 1.05·89-s − 1.82·97-s + 1.99·101-s − 1.91·109-s + 1.31·113-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
\(L(1)\) |
\(\approx\) |
\(2.545974735\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.545974735\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 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
−9.291930041394446121515134862713, −8.297459673524558805021059486547, −7.39175441098936056281221819055, −6.54041133652797620495875328535, −5.85343147872713407555026987911, −5.19827868115262711232652659853, −4.33994590451728047683230101676, −2.84661616844335699138792776248, −2.27579086271573897519647003854, −1.09487650279298634751041561892,
1.09487650279298634751041561892, 2.27579086271573897519647003854, 2.84661616844335699138792776248, 4.33994590451728047683230101676, 5.19827868115262711232652659853, 5.85343147872713407555026987911, 6.54041133652797620495875328535, 7.39175441098936056281221819055, 8.297459673524558805021059486547, 9.291930041394446121515134862713