| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 4·11-s − 12-s + 2·13-s + 16-s − 6·17-s − 18-s − 4·19-s − 4·22-s + 24-s − 2·26-s − 27-s − 8·29-s + 8·31-s − 32-s − 4·33-s + 6·34-s + 36-s + 2·37-s + 4·38-s − 2·39-s + 10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.852·22-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 1.48·29-s + 1.43·31-s − 0.176·32-s − 0.696·33-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.648·38-s − 0.320·39-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.15103705933175, −13.89899017879313, −13.00896887778944, −12.79491814786082, −12.17784207180726, −11.53175854604062, −11.23494495155365, −10.83934227293617, −10.34965265582619, −9.635795297395969, −9.117260063688133, −8.883327964753638, −8.285397558829856, −7.489696840001053, −7.183812030570369, −6.412260620661906, −6.122737887523674, −5.774812130954638, −4.642978099417841, −4.289854538605220, −3.813646421366796, −2.837041479570775, −2.195701697680307, −1.509384993861520, −0.8462108485297128, 0,
0.8462108485297128, 1.509384993861520, 2.195701697680307, 2.837041479570775, 3.813646421366796, 4.289854538605220, 4.642978099417841, 5.774812130954638, 6.122737887523674, 6.412260620661906, 7.183812030570369, 7.489696840001053, 8.285397558829856, 8.883327964753638, 9.117260063688133, 9.635795297395969, 10.34965265582619, 10.83934227293617, 11.23494495155365, 11.53175854604062, 12.17784207180726, 12.79491814786082, 13.00896887778944, 13.89899017879313, 14.15103705933175
