L(s) = 1 | − 5-s + 11-s + i·19-s + (−1 − i)23-s + 25-s + i·29-s + 31-s + (1 − i)37-s − 41-s + (1 + i)43-s + (1 − i)47-s + i·49-s − 55-s + i·59-s + (1 − i)67-s + ⋯ |
L(s) = 1 | − 5-s + 11-s + i·19-s + (−1 − i)23-s + 25-s + i·29-s + 31-s + (1 − i)37-s − 41-s + (1 + i)43-s + (1 − i)47-s + i·49-s − 55-s + i·59-s + (1 − i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.082501713\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.082501713\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 - iT - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 + (-1 + i)T - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (-1 + i)T - iT^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 + (-1 - i)T + iT^{2} \) |
| 89 | \( 1 - iT - T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−8.777362199449354981429506453265, −8.083620340777924822969344459628, −7.49471436028805230598606804052, −6.58514961415288816056875635274, −6.01944252265497369817833712457, −4.85674664331473096631339434902, −4.09223761042493501301846388077, −3.53514929606978651006794279744, −2.36223415565357696474986307221, −1.02092110057465979836534528953,
0.884774419936471199632935260158, 2.29999990034130846099545278391, 3.40691136244679509257435966282, 4.11135242795724782348335774594, 4.78124692557483828853010400450, 5.87215468540369274275750694911, 6.66332884956098550923252036595, 7.32499834147698904851577333336, 8.105924723056489964935689644099, 8.676756732960514877934253253136