| L(s) = 1 | − 3i·9-s + (−5 + 5i)13-s + (5 + 5i)17-s + 4i·29-s + (5 + 5i)37-s + 8·41-s + 7i·49-s + (5 − 5i)53-s + 12·61-s + (−5 + 5i)73-s − 9·81-s + 16i·89-s + (−5 − 5i)97-s − 2·101-s + 6i·109-s + ⋯ |
| L(s) = 1 | − i·9-s + (−1.38 + 1.38i)13-s + (1.21 + 1.21i)17-s + 0.742i·29-s + (0.821 + 0.821i)37-s + 1.24·41-s + i·49-s + (0.686 − 0.686i)53-s + 1.53·61-s + (−0.585 + 0.585i)73-s − 81-s + 1.69i·89-s + (−0.507 − 0.507i)97-s − 0.199·101-s + 0.574i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.433931709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.433931709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3iT^{2} \) |
| 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (5 - 5i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5 - 5i)T + 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (-5 - 5i)T + 37iT^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (-5 + 5i)T - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (5 - 5i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 - 16iT - 89T^{2} \) |
| 97 | \( 1 + (5 + 5i)T + 97iT^{2} \) |
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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
−9.601592281465952282014532857786, −8.851430931675751862171392110095, −7.918244794743921448562385195935, −7.07564203753473218927458952273, −6.38341562420334819823389927008, −5.49081465104499543685072245073, −4.42967311077520984884769525126, −3.66124572424794132344560121879, −2.50266323062434627352120824100, −1.22659962426848624835611065649,
0.61260532860234294807589608905, 2.33298662812018307122173189170, 3.01561917309734725438421389998, 4.37204691372230035009015211978, 5.28335749916440827837260673826, 5.72123793238406465915987156058, 7.21314894823549302892280489246, 7.61040917803321066918598047893, 8.287930654625881369253654658357, 9.500237568668142842893983551188
