| 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·25-s + i·27-s + (−1 + i)29-s + 33-s + i·37-s + i·41-s + 47-s + (1 − i)51-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·25-s + i·27-s + (−1 + i)29-s + 33-s + i·37-s + i·41-s + 47-s + (1 − i)51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6069956411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6069956411\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - iT \) |
| good | 3 | \( 1 - iT - T^{2} \) |
| 5 | \( 1 + iT^{2} \) |
| 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
−13.49148832149289409247800683737, −12.38211825179779823233357558045, −11.23585659445774750887200017992, −10.19320403310866793307351058503, −9.522405592957802838890433163166, −8.444784212317855955232376186194, −6.89050957330215446206173448352, −5.69019913107289534205610998386, −4.27714347876596393041059830264, −3.09928043927083506759952653755,
2.08628502080588857118603629212, 3.95290976747288843243538632860, 5.77021853794458276231113710756, 6.94968605451004120504551546434, 7.56795461850848674019017603429, 9.141281680914141676207393897133, 9.990264392653704431458007765550, 11.37649815088232660667564705338, 12.43625305431146567598223854948, 13.10816457337357686041792780832
