| L(s) = 1 | − 2-s − 3-s + 6-s − 7-s + 8-s + 9-s − 11-s + 13-s + 14-s − 16-s − 2·17-s − 18-s + 21-s + 22-s − 24-s + 25-s − 26-s − 27-s + 31-s + 33-s + 2·34-s − 37-s − 39-s − 42-s + 47-s + 48-s + 49-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 6-s − 7-s + 8-s + 9-s − 11-s + 13-s + 14-s − 16-s − 2·17-s − 18-s + 21-s + 22-s − 24-s + 25-s − 26-s − 27-s + 31-s + 33-s + 2·34-s − 37-s − 39-s − 42-s + 47-s + 48-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3106572366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3106572366\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 + T )^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−9.882135956338356998222681569563, −8.881959414138595301762800402519, −8.451361584982598223417467606620, −7.14759934664058482056151032968, −6.73267332420600299602757377400, −5.73248286360113816909518686473, −4.76651077498915362377686080351, −3.88729086528946771977314854155, −2.31663770145752858291424224513, −0.71353637571973941778247192249,
0.71353637571973941778247192249, 2.31663770145752858291424224513, 3.88729086528946771977314854155, 4.76651077498915362377686080351, 5.73248286360113816909518686473, 6.73267332420600299602757377400, 7.14759934664058482056151032968, 8.451361584982598223417467606620, 8.881959414138595301762800402519, 9.882135956338356998222681569563
