L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + i·9-s + (−0.5 + 0.866i)11-s + 1.00i·14-s + 1.00·16-s + (0.707 − 0.707i)18-s + (0.965 − 0.258i)22-s + (1.22 + 1.22i)23-s + 1.73·29-s + (1.22 − 1.22i)37-s + (−0.707 + 0.707i)43-s − 1.73i·46-s + 1.00i·49-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + i·9-s + (−0.5 + 0.866i)11-s + 1.00i·14-s + 1.00·16-s + (0.707 − 0.707i)18-s + (0.965 − 0.258i)22-s + (1.22 + 1.22i)23-s + 1.73·29-s + (1.22 − 1.22i)37-s + (−0.707 + 0.707i)43-s − 1.73i·46-s + 1.00i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6304867283\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6304867283\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 3 | \( 1 - iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 29 | \( 1 - 1.73T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.590750636608045121463550029526, −8.807129803057370437094631628463, −7.82439911285432407767421210151, −7.25815674562521146960369252099, −6.23986686428447947806892639301, −5.24319515473300988455971933639, −4.48949623889906382995447803777, −3.14747836967154813641013206123, −2.35071810926086239450272373274, −1.16301696950919786600312996674,
0.68376718693344766621465757397, 2.86074608424259575002916647352, 3.24829722997517536025698205217, 4.59863582107155128386829971960, 5.81054437118105139804442000036, 6.45741181010360322582296269897, 6.89473788628772646164827166542, 8.091364174883814501364510159606, 8.605064687381194944070593015379, 9.164745529753125658413340534736