L(s) = 1 | + (0.955 + 0.294i)3-s + (0.0747 + 0.997i)7-s + (0.826 + 0.563i)9-s + (−1.72 − 0.829i)13-s + (−0.0747 − 0.129i)19-s + (−0.222 + 0.974i)21-s + (0.0747 − 0.997i)25-s + (0.623 + 0.781i)27-s + (0.988 − 1.71i)31-s + (−0.722 − 0.108i)37-s + (−1.40 − 1.29i)39-s + (−0.367 + 1.61i)43-s + (−0.988 + 0.149i)49-s + (−0.0332 − 0.145i)57-s + (−1.23 − 0.185i)61-s + ⋯ |
L(s) = 1 | + (0.955 + 0.294i)3-s + (0.0747 + 0.997i)7-s + (0.826 + 0.563i)9-s + (−1.72 − 0.829i)13-s + (−0.0747 − 0.129i)19-s + (−0.222 + 0.974i)21-s + (0.0747 − 0.997i)25-s + (0.623 + 0.781i)27-s + (0.988 − 1.71i)31-s + (−0.722 − 0.108i)37-s + (−1.40 − 1.29i)39-s + (−0.367 + 1.61i)43-s + (−0.988 + 0.149i)49-s + (−0.0332 − 0.145i)57-s + (−1.23 − 0.185i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.180895374\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.180895374\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.955 - 0.294i)T \) |
| 7 | \( 1 + (-0.0747 - 0.997i)T \) |
good | 5 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 11 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 13 | \( 1 + (1.72 + 0.829i)T + (0.623 + 0.781i)T^{2} \) |
| 17 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 19 | \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 29 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 31 | \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.722 + 0.108i)T + (0.955 + 0.294i)T^{2} \) |
| 41 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 43 | \( 1 + (0.367 - 1.61i)T + (-0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 53 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 59 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 61 | \( 1 + (1.23 + 0.185i)T + (0.955 + 0.294i)T^{2} \) |
| 67 | \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.0546 + 0.728i)T + (-0.988 - 0.149i)T^{2} \) |
| 79 | \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 97 | \( 1 + 1.80T + T^{2} \) |
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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
−10.81108296844765077789176212965, −9.788237941107163711228820923545, −9.417986382029737363457824390255, −8.208362343833780376133810583147, −7.80213804058705718923847174302, −6.53160259441540444603463224388, −5.26957849517628938219726786245, −4.43978048812425102550902004829, −2.95331654866925326455233834708, −2.26525933591875023635767300839,
1.67903677522444305254266723655, 2.99851932164108026702505154775, 4.12137799058218943481813301017, 5.06381654486377712037030853294, 6.87950073630855021649770342849, 7.13093244527214949503733897023, 8.149942581575917872035370934960, 9.101983417021709464379179872515, 9.901336312715754622516277214113, 10.59266794104363523535901504593