L(s) = 1 | + 1.41·3-s − 5-s + 4.24·7-s − 0.999·9-s + 2.82·11-s + 6·13-s − 1.41·15-s − 6·17-s + 6·21-s + 4.24·23-s + 25-s − 5.65·27-s − 8.48·31-s + 4.00·33-s − 4.24·35-s + 6·37-s + 8.48·39-s + 4.24·43-s + 0.999·45-s + 4.24·47-s + 10.9·49-s − 8.48·51-s + 6·53-s − 2.82·55-s − 11.3·59-s + 6·61-s − 4.24·63-s + ⋯ |
L(s) = 1 | + 0.816·3-s − 0.447·5-s + 1.60·7-s − 0.333·9-s + 0.852·11-s + 1.66·13-s − 0.365·15-s − 1.45·17-s + 1.30·21-s + 0.884·23-s + 0.200·25-s − 1.08·27-s − 1.52·31-s + 0.696·33-s − 0.717·35-s + 0.986·37-s + 1.35·39-s + 0.646·43-s + 0.149·45-s + 0.618·47-s + 1.57·49-s − 1.18·51-s + 0.824·53-s − 0.381·55-s − 1.47·59-s + 0.768·61-s − 0.534·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.513723464\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.513723464\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4.24T + 43T^{2} \) |
| 47 | \( 1 - 4.24T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.135738007135489841500532466931, −8.872462334402988355068017525881, −8.206639157502084528486600815541, −7.45821065340662889937664742530, −6.43270933122789841211637865148, −5.39347463419222355222088833077, −4.28205376380185187932448349488, −3.68457874219985765989446656655, −2.35306015800440012562474726032, −1.28485458580888682346600691311,
1.28485458580888682346600691311, 2.35306015800440012562474726032, 3.68457874219985765989446656655, 4.28205376380185187932448349488, 5.39347463419222355222088833077, 6.43270933122789841211637865148, 7.45821065340662889937664742530, 8.206639157502084528486600815541, 8.872462334402988355068017525881, 9.135738007135489841500532466931