L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + 5-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)10-s − 1.41i·11-s − 1.00·14-s − 1.00·16-s + (−1 + i)17-s + 1.41i·19-s − 1.00i·20-s + (−1.00 − 1.00i)22-s + 25-s + (−0.707 + 0.707i)28-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + 5-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)10-s − 1.41i·11-s − 1.00·14-s − 1.00·16-s + (−1 + i)17-s + 1.41i·19-s − 1.00i·20-s + (−1.00 − 1.00i)22-s + 25-s + (−0.707 + 0.707i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.594554252\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.594554252\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (1 - i)T - iT^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 + 2iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 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
−9.985060706171560795946849417372, −9.059483063031849444247540840576, −8.226702133397213320268038174334, −6.70837816087971111233251865946, −6.16611103227721296195916685166, −5.55880597242699760921462607712, −4.28167444796363657382045257402, −3.49025780032061954018839079916, −2.48777114432843828978640099774, −1.19643705564035640918987382617,
2.33572953309973904024194901449, 2.85456487422485691404274893742, 4.53352003046742537680602568600, 4.96310940970327705418246472101, 6.09118537405788621281013089458, 6.66352544892860029015835683131, 7.32089807297289391367550641204, 8.538479576660980878176457266066, 9.420743069413805176751368100991, 9.668279572763449269209909433046