L(s) = 1 | + 2.41·2-s + 3.82·4-s + 5-s + 2·7-s + 4.41·8-s + 2.41·10-s + 1.17·13-s + 4.82·14-s + 2.99·16-s + 6.82·17-s + 3.82·20-s + 2.82·23-s + 25-s + 2.82·26-s + 7.65·28-s − 3.65·29-s − 1.58·32-s + 16.4·34-s + 2·35-s − 7.65·37-s + 4.41·40-s + 6·41-s + 6·43-s + 6.82·46-s − 2.82·47-s − 3·49-s + 2.41·50-s + ⋯ |
L(s) = 1 | + 1.70·2-s + 1.91·4-s + 0.447·5-s + 0.755·7-s + 1.56·8-s + 0.763·10-s + 0.324·13-s + 1.29·14-s + 0.749·16-s + 1.65·17-s + 0.856·20-s + 0.589·23-s + 0.200·25-s + 0.554·26-s + 1.44·28-s − 0.679·29-s − 0.280·32-s + 2.82·34-s + 0.338·35-s − 1.25·37-s + 0.697·40-s + 0.937·41-s + 0.914·43-s + 1.00·46-s − 0.412·47-s − 0.428·49-s + 0.341·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.358701335\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.358701335\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 1.17T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−7.85954057968084156005808612198, −7.30097239134651260879117695149, −6.43440016195920616082350539486, −5.75213654005785566336985750282, −5.23856763699651031759734012370, −4.64144588106317236533705950964, −3.69795996150534809220635861927, −3.12889975732287872040313663241, −2.13364715184755587713038158870, −1.26734886930746115142007972380,
1.26734886930746115142007972380, 2.13364715184755587713038158870, 3.12889975732287872040313663241, 3.69795996150534809220635861927, 4.64144588106317236533705950964, 5.23856763699651031759734012370, 5.75213654005785566336985750282, 6.43440016195920616082350539486, 7.30097239134651260879117695149, 7.85954057968084156005808612198