L(s) = 1 | + 3-s − 5-s + 9-s − 2·13-s − 15-s + 6·17-s + 8·19-s + 25-s + 27-s + 6·29-s − 4·31-s − 10·37-s − 2·39-s + 6·41-s + 4·43-s − 45-s + 6·51-s − 6·53-s + 8·57-s − 12·59-s + 10·61-s + 2·65-s + 4·67-s − 12·71-s + 10·73-s + 75-s − 8·79-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 1.45·17-s + 1.83·19-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 1.64·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s + 0.840·51-s − 0.824·53-s + 1.05·57-s − 1.56·59-s + 1.28·61-s + 0.248·65-s + 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.115·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.668421033\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.668421033\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.14828930060386, −15.93617702121665, −15.32125939879068, −14.51394661466724, −14.20288657539311, −13.80774328663217, −12.91392023252724, −12.30463122639478, −11.99242229624512, −11.30632343239036, −10.47226208397189, −9.997083055998300, −9.367965458919266, −8.854820418843128, −7.960593301944536, −7.606746052002103, −7.130108824311213, −6.225216221334624, −5.328568460355334, −4.930511681215701, −3.911281628220572, −3.289204483716040, −2.764629871397258, −1.625349451864508, −0.7591250888932235,
0.7591250888932235, 1.625349451864508, 2.764629871397258, 3.289204483716040, 3.911281628220572, 4.930511681215701, 5.328568460355334, 6.225216221334624, 7.130108824311213, 7.606746052002103, 7.960593301944536, 8.854820418843128, 9.367965458919266, 9.997083055998300, 10.47226208397189, 11.30632343239036, 11.99242229624512, 12.30463122639478, 12.91392023252724, 13.80774328663217, 14.20288657539311, 14.51394661466724, 15.32125939879068, 15.93617702121665, 16.14828930060386