L(s) = 1 | + 2.44·3-s − 7-s + 2.99·9-s − 4.89·11-s + 0.449·13-s − 2·17-s − 6.44·19-s − 2.44·21-s − 6.89·23-s − 2.89·29-s + 0.898·31-s − 11.9·33-s − 2·37-s + 1.10·39-s − 10.8·41-s + 8.89·43-s − 0.898·47-s + 49-s − 4.89·51-s + 1.10·53-s − 15.7·57-s + 6.44·59-s + 8.44·61-s − 2.99·63-s − 8·67-s − 16.8·69-s + 10.8·71-s + ⋯ |
L(s) = 1 | + 1.41·3-s − 0.377·7-s + 0.999·9-s − 1.47·11-s + 0.124·13-s − 0.485·17-s − 1.47·19-s − 0.534·21-s − 1.43·23-s − 0.538·29-s + 0.161·31-s − 2.08·33-s − 0.328·37-s + 0.176·39-s − 1.70·41-s + 1.35·43-s − 0.131·47-s + 0.142·49-s − 0.685·51-s + 0.151·53-s − 2.09·57-s + 0.839·59-s + 1.08·61-s − 0.377·63-s − 0.977·67-s − 2.03·69-s + 1.29·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 0.449T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 23 | \( 1 + 6.89T + 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 - 0.898T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 8.89T + 43T^{2} \) |
| 47 | \( 1 + 0.898T + 47T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 - 8.44T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 6.89T + 73T^{2} \) |
| 79 | \( 1 - 2.89T + 79T^{2} \) |
| 83 | \( 1 - 2.44T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 3.79T + 97T^{2} \) |
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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
−8.346135390691261229146037556740, −7.957917559360360332292014055983, −7.09469205153947435410375484788, −6.21144180293594010690559585904, −5.27693023530872518385519657807, −4.20105028102060935871931099403, −3.52280060844979710452524170987, −2.48647846848973479442346877592, −2.05148822211094492754286695184, 0,
2.05148822211094492754286695184, 2.48647846848973479442346877592, 3.52280060844979710452524170987, 4.20105028102060935871931099403, 5.27693023530872518385519657807, 6.21144180293594010690559585904, 7.09469205153947435410375484788, 7.957917559360360332292014055983, 8.346135390691261229146037556740