L(s) = 1 | + (0.453 − 0.891i)5-s + (−1.76 + 0.278i)13-s + (−1.44 − 0.734i)17-s + (−0.587 − 0.809i)25-s + (0.610 − 1.87i)29-s + (−0.309 − 1.95i)37-s + (1.04 − 1.44i)41-s + i·49-s + (0.550 − 0.280i)53-s + (−1.53 + 1.11i)61-s + (−0.550 + 1.69i)65-s + (−0.142 + 0.896i)73-s + (−1.30 + 0.951i)85-s + (0.253 − 0.183i)89-s + (0.278 − 0.142i)97-s + ⋯ |
L(s) = 1 | + (0.453 − 0.891i)5-s + (−1.76 + 0.278i)13-s + (−1.44 − 0.734i)17-s + (−0.587 − 0.809i)25-s + (0.610 − 1.87i)29-s + (−0.309 − 1.95i)37-s + (1.04 − 1.44i)41-s + i·49-s + (0.550 − 0.280i)53-s + (−1.53 + 1.11i)61-s + (−0.550 + 1.69i)65-s + (−0.142 + 0.896i)73-s + (−1.30 + 0.951i)85-s + (0.253 − 0.183i)89-s + (0.278 − 0.142i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8905316982\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8905316982\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.453 + 0.891i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (1.76 - 0.278i)T + (0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (1.44 + 0.734i)T + (0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 29 | \( 1 + (-0.610 + 1.87i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 + 1.95i)T + (-0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (-1.04 + 1.44i)T + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (-0.550 + 0.280i)T + (0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + (-0.253 + 0.183i)T + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \) |
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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
−8.695287591377631609399346915317, −7.62281368235983371454368696716, −7.17147366049323127698254927435, −6.14877518569904793487118096018, −5.44615129203995068065571105517, −4.56633723816015888289074031857, −4.20351873810532290775074661892, −2.53642790460444296731846135151, −2.13065084744309204465015778700, −0.47229230530386823641710718887,
1.71343137067215097141543287452, 2.62630468496071689279350756523, 3.28829491816249317757726074935, 4.56172684434302265735027423898, 5.08889733574927997881670659161, 6.20584563981012636716119451037, 6.72455572414993930062131187141, 7.37062800364181023750817936796, 8.188694815618362208470563307698, 9.046670518998777384373851842203