L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 7-s + 0.999·8-s − 9-s + 1.73i·11-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)18-s + (1.49 − 0.866i)22-s + 23-s + (−0.499 + 0.866i)28-s + 1.73i·29-s + (−0.499 + 0.866i)32-s + (0.499 − 0.866i)36-s + 1.73i·37-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 7-s + 0.999·8-s − 9-s + 1.73i·11-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)18-s + (1.49 − 0.866i)22-s + 23-s + (−0.499 + 0.866i)28-s + 1.73i·29-s + (−0.499 + 0.866i)32-s + (0.499 − 0.866i)36-s + 1.73i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8170917516\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8170917516\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + T^{2} \) |
| 11 | \( 1 - 1.73iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - 1.73iT - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.73iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.73iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.73iT - T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.779024569338101610645830766425, −8.974190698158375424781899021951, −8.374357162294601641240779888785, −7.51506998379007194065014130210, −6.80216194492556374663850481879, −5.14470689008525013489662826487, −4.77125087247963427399857683850, −3.52479085271814207905637985186, −2.43350354613194614869655619725, −1.49688086274837542164910117526,
0.890299391740152946017137073877, 2.53036363345507421042197738697, 3.90274245315059242989351925157, 5.06511350778806645121820480257, 5.74418343413207275652897321280, 6.36151742882763868532942070243, 7.59126366490503593951134228904, 8.175960017039092695013840151809, 8.752864014611837125206330724510, 9.420855792220089780829821102286