L(s) = 1 | + 2.61·3-s − 1.63·5-s + 3.46·7-s + 3.85·9-s + 1.65·11-s − 2.82·13-s − 4.27·15-s + 6.37·17-s + 3.98·19-s + 9.07·21-s − 2.04·23-s − 2.33·25-s + 2.25·27-s − 0.0143·29-s + 7.31·31-s + 4.32·33-s − 5.65·35-s − 4.31·37-s − 7.38·39-s + 10.1·41-s − 1.72·43-s − 6.29·45-s − 47-s + 5.00·49-s + 16.6·51-s + 5.63·53-s − 2.69·55-s + ⋯ |
L(s) = 1 | + 1.51·3-s − 0.729·5-s + 1.30·7-s + 1.28·9-s + 0.498·11-s − 0.782·13-s − 1.10·15-s + 1.54·17-s + 0.915·19-s + 1.98·21-s − 0.426·23-s − 0.467·25-s + 0.433·27-s − 0.00265·29-s + 1.31·31-s + 0.753·33-s − 0.955·35-s − 0.709·37-s − 1.18·39-s + 1.58·41-s − 0.263·43-s − 0.938·45-s − 0.145·47-s + 0.715·49-s + 2.33·51-s + 0.774·53-s − 0.363·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.996124197\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.996124197\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 47 | \( 1 + T \) |
good | 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 + 1.63T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 6.37T + 17T^{2} \) |
| 19 | \( 1 - 3.98T + 19T^{2} \) |
| 23 | \( 1 + 2.04T + 23T^{2} \) |
| 29 | \( 1 + 0.0143T + 29T^{2} \) |
| 31 | \( 1 - 7.31T + 31T^{2} \) |
| 37 | \( 1 + 4.31T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 1.72T + 43T^{2} \) |
| 53 | \( 1 - 5.63T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 1.84T + 61T^{2} \) |
| 67 | \( 1 - 3.19T + 67T^{2} \) |
| 71 | \( 1 + 1.78T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 7.77T + 83T^{2} \) |
| 89 | \( 1 + 0.300T + 89T^{2} \) |
| 97 | \( 1 - 3.62T + 97T^{2} \) |
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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
−7.997941641356191025845373255754, −7.68435554437467981265582664939, −7.12811705742385968389568786127, −5.86811520899257565668191422995, −4.99052992526068317701730435807, −4.27206794506020826555738708359, −3.57569974946720765291269438427, −2.86002246953585681065948619165, −1.94674055022828562120700970610, −1.05158084530840030377964285587,
1.05158084530840030377964285587, 1.94674055022828562120700970610, 2.86002246953585681065948619165, 3.57569974946720765291269438427, 4.27206794506020826555738708359, 4.99052992526068317701730435807, 5.86811520899257565668191422995, 7.12811705742385968389568786127, 7.68435554437467981265582664939, 7.997941641356191025845373255754