L(s) = 1 | + (0.860 − 0.860i)2-s − 0.482i·4-s + (0.608 + 0.793i)5-s + (0.445 + 0.445i)8-s + (1.20 + 0.158i)10-s − 0.765i·11-s + (−0.707 + 0.707i)13-s + 1.24·16-s + (0.382 − 0.293i)20-s + (−0.658 − 0.658i)22-s + (−0.258 + 0.965i)25-s + 1.21i·26-s + (0.630 − 0.630i)32-s + (−0.0822 + 0.624i)40-s − 1.84i·41-s + ⋯ |
L(s) = 1 | + (0.860 − 0.860i)2-s − 0.482i·4-s + (0.608 + 0.793i)5-s + (0.445 + 0.445i)8-s + (1.20 + 0.158i)10-s − 0.765i·11-s + (−0.707 + 0.707i)13-s + 1.24·16-s + (0.382 − 0.293i)20-s + (−0.658 − 0.658i)22-s + (−0.258 + 0.965i)25-s + 1.21i·26-s + (0.630 − 0.630i)32-s + (−0.0822 + 0.624i)40-s − 1.84i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.976379440\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.976379440\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.608 - 0.793i)T \) |
| 13 | \( 1 + (0.707 - 0.707i)T \) |
good | 2 | \( 1 + (-0.860 + 0.860i)T - iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + 0.765iT - T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + 1.84iT - T^{2} \) |
| 43 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 47 | \( 1 + (-1.12 + 1.12i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - 0.261T + T^{2} \) |
| 61 | \( 1 + 1.93T + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.58iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (-0.184 - 0.184i)T + iT^{2} \) |
| 89 | \( 1 + 1.98T + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.616137694193932725454373848741, −8.852285263427887165682527193572, −7.74570554926791551713264967093, −6.99359141711010862673817353090, −6.01784759878842613409551871949, −5.28941837355827788431449181226, −4.30403495990947505499187530028, −3.40625719191114367522039562647, −2.61745362105258369284024065838, −1.76534833526587080631205566889,
1.37641259683929665212270201746, 2.72300084096474299505063825070, 4.12685526180233811924501141799, 4.80920340215690177300347216752, 5.43875255815047401819553432056, 6.15989438456326051230104769440, 7.03226277076284229903998023026, 7.76032857710566221301552037962, 8.585288217400419781561342222340, 9.771490705497110672374598562013