| L(s) = 1 | + 2.27·2-s + 3-s + 3.15·4-s − 5-s + 2.27·6-s + 2.63·8-s + 9-s − 2.27·10-s + 5.05·11-s + 3.15·12-s + 0.224·13-s − 15-s − 0.338·16-s − 3.59·17-s + 2.27·18-s + 4.78·19-s − 3.15·20-s + 11.4·22-s + 7.01·23-s + 2.63·24-s + 25-s + 0.510·26-s + 27-s − 8.74·29-s − 2.27·30-s − 8.78·31-s − 6.03·32-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 0.577·3-s + 1.57·4-s − 0.447·5-s + 0.927·6-s + 0.930·8-s + 0.333·9-s − 0.718·10-s + 1.52·11-s + 0.911·12-s + 0.0623·13-s − 0.258·15-s − 0.0847·16-s − 0.872·17-s + 0.535·18-s + 1.09·19-s − 0.706·20-s + 2.44·22-s + 1.46·23-s + 0.537·24-s + 0.200·25-s + 0.100·26-s + 0.192·27-s − 1.62·29-s − 0.414·30-s − 1.57·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.230445553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.230445553\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.27T + 2T^{2} \) |
| 11 | \( 1 - 5.05T + 11T^{2} \) |
| 13 | \( 1 - 0.224T + 13T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 19 | \( 1 - 4.78T + 19T^{2} \) |
| 23 | \( 1 - 7.01T + 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 + 8.78T + 31T^{2} \) |
| 37 | \( 1 + 5.91T + 37T^{2} \) |
| 41 | \( 1 + 8.42T + 41T^{2} \) |
| 43 | \( 1 + 4.31T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 0.361T + 53T^{2} \) |
| 59 | \( 1 - 6.25T + 59T^{2} \) |
| 61 | \( 1 + 6.58T + 61T^{2} \) |
| 67 | \( 1 + 8.36T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 4.86T + 73T^{2} \) |
| 79 | \( 1 - 2.88T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 + 2.48T + 97T^{2} \) |
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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
−10.78463197841359729062494154341, −9.254604605696625200299545463010, −8.869501231962254204296023652445, −7.27789265118391045525118253625, −6.91068251843255676469212814361, −5.71550388704686528972797044888, −4.76375665097395989716430818760, −3.77730702638999134541527639441, −3.27802794638541044011758408203, −1.79182185636844248975787007044,
1.79182185636844248975787007044, 3.27802794638541044011758408203, 3.77730702638999134541527639441, 4.76375665097395989716430818760, 5.71550388704686528972797044888, 6.91068251843255676469212814361, 7.27789265118391045525118253625, 8.869501231962254204296023652445, 9.254604605696625200299545463010, 10.78463197841359729062494154341
