L(s) = 1 | + (−0.587 − 0.809i)9-s + (0.142 − 0.896i)13-s + (−0.278 + 0.142i)17-s + (1.11 − 0.363i)29-s + (−1.95 − 0.309i)37-s + (1.53 − 1.11i)41-s − i·49-s + (−1.58 − 0.809i)53-s + (−1.53 − 1.11i)61-s + (1.76 − 0.278i)73-s + (−0.309 + 0.951i)81-s + (0.690 − 0.951i)89-s + (−0.896 + 1.76i)97-s + 1.61·101-s + (0.363 + 0.5i)109-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.809i)9-s + (0.142 − 0.896i)13-s + (−0.278 + 0.142i)17-s + (1.11 − 0.363i)29-s + (−1.95 − 0.309i)37-s + (1.53 − 1.11i)41-s − i·49-s + (−1.58 − 0.809i)53-s + (−1.53 − 1.11i)61-s + (1.76 − 0.278i)73-s + (−0.309 + 0.951i)81-s + (0.690 − 0.951i)89-s + (−0.896 + 1.76i)97-s + 1.61·101-s + (0.363 + 0.5i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.026883581\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.026883581\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 29 | \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (1.95 + 0.309i)T + (0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (1.58 + 0.809i)T + (0.587 + 0.809i)T^{2} \) |
| 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 | \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
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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
−8.453743268200076429784660292229, −7.84848210786927213243968838719, −6.89533397988922058432011917778, −6.25662276746199384674519340248, −5.53936783410052730666600455816, −4.75299605544169250106007051307, −3.67651181652074372227123776237, −3.09868494906398745957963937224, −2.00469962958422626197963829526, −0.57895258705990064752217201139,
1.43870518205962038865426797308, 2.46867510874005352708696823966, 3.27606640128429834635803549029, 4.44697136689052723308720081412, 4.91242802257040244233212988725, 5.92422512567437651674502343991, 6.54302245244872140342536079358, 7.39956522403477148091255275728, 8.074462164533178584142168928965, 8.823471277066082851244550107488