L(s) = 1 | + 2-s + 4-s + (−0.707 − 0.707i)7-s + 8-s + (0.707 − 0.292i)11-s + (−0.707 − 0.707i)14-s + 16-s + (0.707 − 0.292i)22-s + (0.707 − 0.707i)25-s + (−0.707 − 0.707i)28-s + (−0.707 + 1.70i)29-s + 32-s + (−0.292 − 0.707i)37-s + (−0.707 − 1.70i)43-s + (0.707 − 0.292i)44-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + (−0.707 − 0.707i)7-s + 8-s + (0.707 − 0.292i)11-s + (−0.707 − 0.707i)14-s + 16-s + (0.707 − 0.292i)22-s + (0.707 − 0.707i)25-s + (−0.707 − 0.707i)28-s + (−0.707 + 1.70i)29-s + 32-s + (−0.292 − 0.707i)37-s + (−0.707 − 1.70i)43-s + (0.707 − 0.292i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.158270120\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.158270120\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 5 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 + (1 - i)T - iT^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.284440489113700431419774130126, −8.523245711491623987210102476597, −7.25224170970238946719627496066, −6.97467575411645836447957178940, −6.07447399905347671047239071367, −5.29542797413871424001298801107, −4.23431157981915165963925060638, −3.62238035867631709600739697164, −2.73522286090890548931560528823, −1.34491021891440770966109071727,
1.68764139334497581913424624736, 2.77389108253624107216387536119, 3.57928445123212443906186708855, 4.50485158055025781113533528055, 5.39104287709016801832442784285, 6.21090905980475640858097136447, 6.73286939350955501460688611893, 7.63949053865133953819412636171, 8.573656742177748377868043290133, 9.536258721331738727696632878067