L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s − 7-s + 9-s + 5·11-s − 2·12-s + 4·13-s − 2·14-s − 4·16-s − 6·17-s + 2·18-s + 5·19-s + 21-s + 10·22-s + 8·26-s − 27-s − 2·28-s + 6·29-s + 5·31-s − 8·32-s − 5·33-s − 12·34-s + 2·36-s − 6·37-s + 10·38-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s − 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.577·12-s + 1.10·13-s − 0.534·14-s − 16-s − 1.45·17-s + 0.471·18-s + 1.14·19-s + 0.218·21-s + 2.13·22-s + 1.56·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s + 0.898·31-s − 1.41·32-s − 0.870·33-s − 2.05·34-s + 1/3·36-s − 0.986·37-s + 1.62·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.167940975\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.167940975\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.76483506221950, −12.25513307743283, −11.92751969219832, −11.60194312386013, −11.08118056911120, −10.77042649067577, −10.08082298111034, −9.476163128979712, −9.103289837206024, −8.656566075353458, −8.135812008138050, −7.225544034999640, −6.816550396125532, −6.398830612363581, −6.236121827477341, −5.589522877835202, −5.080921153027842, −4.529327754811790, −4.109294349690908, −3.724107225578473, −3.147507211493669, −2.634879748181131, −1.803661635786844, −1.208278055666939, −0.5211696794285715,
0.5211696794285715, 1.208278055666939, 1.803661635786844, 2.634879748181131, 3.147507211493669, 3.724107225578473, 4.109294349690908, 4.529327754811790, 5.080921153027842, 5.589522877835202, 6.236121827477341, 6.398830612363581, 6.816550396125532, 7.225544034999640, 8.135812008138050, 8.656566075353458, 9.103289837206024, 9.476163128979712, 10.08082298111034, 10.77042649067577, 11.08118056911120, 11.60194312386013, 11.92751969219832, 12.25513307743283, 12.76483506221950