L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−1.73 − 1.41i)5-s + (2 + 2i)7-s + (−0.707 − 0.707i)8-s + (−2.22 + 0.224i)10-s + 2.82i·11-s + (−2.44 + 2.44i)13-s + 2.82·14-s − 1.00·16-s + (−0.707 + 0.707i)17-s + 6.89i·19-s + (−1.41 + 1.73i)20-s + (2.00 + 2.00i)22-s + (−1.73 − 1.73i)23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.774 − 0.632i)5-s + (0.755 + 0.755i)7-s + (−0.250 − 0.250i)8-s + (−0.703 + 0.0710i)10-s + 0.852i·11-s + (−0.679 + 0.679i)13-s + 0.755·14-s − 0.250·16-s + (−0.171 + 0.171i)17-s + 1.58i·19-s + (−0.316 + 0.387i)20-s + (0.426 + 0.426i)22-s + (−0.361 − 0.361i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.494739087\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.494739087\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.73 + 1.41i)T \) |
| 17 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-2 - 2i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (2.44 - 2.44i)T - 13iT^{2} \) |
| 19 | \( 1 - 6.89iT - 19T^{2} \) |
| 23 | \( 1 + (1.73 + 1.73i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 + (-6.44 - 6.44i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.635iT - 41T^{2} \) |
| 43 | \( 1 + (-3.44 + 3.44i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.82 - 2.82i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.51 - 5.51i)T + 53iT^{2} \) |
| 59 | \( 1 + 4.87T + 59T^{2} \) |
| 61 | \( 1 - 5.34T + 61T^{2} \) |
| 67 | \( 1 + (1.44 + 1.44i)T + 67iT^{2} \) |
| 71 | \( 1 - 0.635iT - 71T^{2} \) |
| 73 | \( 1 + (3.34 - 3.34i)T - 73iT^{2} \) |
| 79 | \( 1 - 6.44iT - 79T^{2} \) |
| 83 | \( 1 + (2.82 + 2.82i)T + 83iT^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + (-6.44 - 6.44i)T + 97iT^{2} \) |
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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
−9.584658036155045782221317908734, −8.795784762013264989861261953652, −8.013633284439700681777813717632, −7.27038723079761419461666252130, −6.09179072299359877631707556779, −5.16467205095176607802619120232, −4.50729463155520036799807163968, −3.78465500037897771961575105867, −2.36867059841804845726070113965, −1.49900257251054669497691355537,
0.50126211814377785633413036639, 2.52234862576925298677096956298, 3.45860417115847832449648167957, 4.35063502903803393760444165213, 5.11005178387508211364270229489, 6.12919492510406968682823876647, 7.15542207215945129678413987311, 7.53245758827933130361213894399, 8.274465317474258844717832492147, 9.174891728186024795550575755881