L(s) = 1 | − i·3-s + 7-s − i·11-s + (1 + i)17-s + (1 + i)19-s − i·21-s + (1 + i)23-s − i·27-s + (−1 + i)29-s − 33-s − i·37-s + i·41-s − 47-s + (1 − i)51-s − 53-s + ⋯ |
L(s) = 1 | − i·3-s + 7-s − i·11-s + (1 + i)17-s + (1 + i)19-s − i·21-s + (1 + i)23-s − i·27-s + (−1 + i)29-s − 33-s − i·37-s + i·41-s − 47-s + (1 − i)51-s − 53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.609501215\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.609501215\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
good | 3 | \( 1 + iT - T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-1 - i)T + iT^{2} \) |
| 19 | \( 1 + (-1 - i)T + iT^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 29 | \( 1 + (1 - i)T - iT^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 41 | \( 1 - iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 + (1 + i)T + iT^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + (1 - i)T - iT^{2} \) |
| 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
−8.278530297918887055281650979792, −7.82784466679219195638833579332, −7.36352551910404613276500251614, −6.37647011753488254974639209808, −5.64531303540702486790841932689, −5.05966832635472737334370281371, −3.79079293939286624844046370711, −3.11994087108081785672116673651, −1.61827613678575879877872910703, −1.31929421655682721656076924638,
1.25082275505334910550519481298, 2.48567393512262704333019007315, 3.43369668097751419319834170452, 4.44911323524444647214184036062, 4.93846139880544497394701628179, 5.39425221866347838874920490589, 6.76955636791342951008482835169, 7.36497555774530328737722817188, 8.042577099153215325023264020322, 8.986853217855073298170146806100