L(s) = 1 | + (0.382 − 0.923i)3-s + (0.923 + 0.382i)4-s + (0.785 + 0.785i)7-s + (−0.707 − 0.707i)9-s + (0.707 − 0.707i)12-s + (−1.83 + 0.555i)13-s + (0.707 + 0.707i)16-s + (−1.26 − 0.124i)19-s + (1.02 − 0.425i)21-s + (−0.980 − 0.195i)25-s + (−0.923 + 0.382i)27-s + (0.425 + 1.02i)28-s + (0.360 − 1.81i)31-s + (−0.382 − 0.923i)36-s + (0.831 − 1.55i)37-s + ⋯ |
L(s) = 1 | + (0.382 − 0.923i)3-s + (0.923 + 0.382i)4-s + (0.785 + 0.785i)7-s + (−0.707 − 0.707i)9-s + (0.707 − 0.707i)12-s + (−1.83 + 0.555i)13-s + (0.707 + 0.707i)16-s + (−1.26 − 0.124i)19-s + (1.02 − 0.425i)21-s + (−0.980 − 0.195i)25-s + (−0.923 + 0.382i)27-s + (0.425 + 1.02i)28-s + (0.360 − 1.81i)31-s + (−0.382 − 0.923i)36-s + (0.831 − 1.55i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.193236339\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.193236339\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.382 + 0.923i)T \) |
| 193 | \( 1 + (-0.923 - 0.382i)T \) |
good | 2 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 5 | \( 1 + (0.980 + 0.195i)T^{2} \) |
| 7 | \( 1 + (-0.785 - 0.785i)T + iT^{2} \) |
| 11 | \( 1 + (-0.195 - 0.980i)T^{2} \) |
| 13 | \( 1 + (1.83 - 0.555i)T + (0.831 - 0.555i)T^{2} \) |
| 17 | \( 1 + (0.980 - 0.195i)T^{2} \) |
| 19 | \( 1 + (1.26 + 0.124i)T + (0.980 + 0.195i)T^{2} \) |
| 23 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 29 | \( 1 + (0.555 - 0.831i)T^{2} \) |
| 31 | \( 1 + (-0.360 + 1.81i)T + (-0.923 - 0.382i)T^{2} \) |
| 37 | \( 1 + (-0.831 + 1.55i)T + (-0.555 - 0.831i)T^{2} \) |
| 41 | \( 1 + (-0.195 - 0.980i)T^{2} \) |
| 43 | \( 1 + (0.275 - 0.275i)T - iT^{2} \) |
| 47 | \( 1 + (0.831 - 0.555i)T^{2} \) |
| 53 | \( 1 + (-0.831 + 0.555i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.195 + 0.0192i)T + (0.980 - 0.195i)T^{2} \) |
| 67 | \( 1 + (0.382 - 1.92i)T + (-0.923 - 0.382i)T^{2} \) |
| 71 | \( 1 + (-0.980 - 0.195i)T^{2} \) |
| 73 | \( 1 + (-0.938 - 1.75i)T + (-0.555 + 0.831i)T^{2} \) |
| 79 | \( 1 + (-0.598 + 0.728i)T + (-0.195 - 0.980i)T^{2} \) |
| 83 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 89 | \( 1 + (-0.831 - 0.555i)T^{2} \) |
| 97 | \( 1 + (-1.38 + 0.275i)T + (0.923 - 0.382i)T^{2} \) |
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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
−11.33513345773501539446855280256, −9.946229917013414090961668171627, −8.925591323397484351612560545858, −7.972145143595714235980692605146, −7.46954149467975515084551961290, −6.50788422789041034887648099902, −5.61408587762114590551532711463, −4.16018230307524136011547803808, −2.36062547143334541703062016245, −2.21027825867868094838694125680,
2.00612519443209139350884772468, 3.14008141214671876647346594448, 4.54280033678521130184425017380, 5.18989724564568915021220570354, 6.49879250002570248333756845334, 7.58976366365501653661722212230, 8.163800258876048372032151227860, 9.502087501025703921029005323812, 10.31495380698388634952432455110, 10.67207631105769072877135369155