L(s) = 1 | + (0.341 − 1.37i)2-s + 2.41i·3-s + (−1.76 − 0.938i)4-s + i·5-s + (3.31 + 0.827i)6-s − 2.44·7-s + (−1.89 + 2.10i)8-s − 2.85·9-s + (1.37 + 0.341i)10-s − 2.59·11-s + (2.26 − 4.27i)12-s − 0.482·13-s + (−0.834 + 3.34i)14-s − 2.41·15-s + (2.23 + 3.31i)16-s − 1.74i·17-s + ⋯ |
L(s) = 1 | + (0.241 − 0.970i)2-s + 1.39i·3-s + (−0.883 − 0.469i)4-s + 0.447i·5-s + (1.35 + 0.337i)6-s − 0.922·7-s + (−0.668 + 0.743i)8-s − 0.950·9-s + (0.433 + 0.108i)10-s − 0.780·11-s + (0.655 − 1.23i)12-s − 0.133·13-s + (−0.223 + 0.895i)14-s − 0.624·15-s + (0.559 + 0.828i)16-s − 0.422i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.224598 + 0.490808i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.224598 + 0.490808i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.341 + 1.37i)T \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (1.73 - 4.47i)T \) |
good | 3 | \( 1 - 2.41iT - 3T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 2.59T + 11T^{2} \) |
| 13 | \( 1 + 0.482T + 13T^{2} \) |
| 17 | \( 1 + 1.74iT - 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 29 | \( 1 - 1.03T + 29T^{2} \) |
| 31 | \( 1 + 2.87iT - 31T^{2} \) |
| 37 | \( 1 - 6.76iT - 37T^{2} \) |
| 41 | \( 1 - 0.907T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 - 4.40iT - 47T^{2} \) |
| 53 | \( 1 - 1.89iT - 53T^{2} \) |
| 59 | \( 1 - 3.79iT - 59T^{2} \) |
| 61 | \( 1 - 8.17iT - 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 + 11.0iT - 71T^{2} \) |
| 73 | \( 1 - 5.06T + 73T^{2} \) |
| 79 | \( 1 - 9.79T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 2.55iT - 89T^{2} \) |
| 97 | \( 1 + 1.34iT - 97T^{2} \) |
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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
−11.02420304934137194289847356644, −10.52346238841751584988814784502, −9.779677325917256222712300521524, −9.258283494522527460585775461199, −8.076024620113252168622198814137, −6.43864641874791507925420056053, −5.35780150017323091775312192458, −4.36028907619154586717831874913, −3.48403778509316616609668232973, −2.52293605881037229796099590336,
0.29123877084114491964748717688, 2.37949745958730153628759959948, 3.97256323094102570682544753545, 5.30980361499263513684824688445, 6.40335038658648746269910735488, 6.78215902666175744934288834358, 7.976673666735738568667411027738, 8.433126536064533361191237480729, 9.563490798192494203105839186034, 10.69460449196800306483619025215