L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 3·11-s − 12-s + 4·13-s + 14-s + 16-s − 18-s + 21-s + 3·22-s − 9·23-s + 24-s − 4·26-s − 27-s − 28-s + 3·29-s + 2·31-s − 32-s + 3·33-s + 36-s + 10·37-s − 4·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.218·21-s + 0.639·22-s − 1.87·23-s + 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.188·28-s + 0.557·29-s + 0.359·31-s − 0.176·32-s + 0.522·33-s + 1/6·36-s + 1.64·37-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 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 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.81502649820538, −11.99257100011101, −11.76704423518807, −11.37227796139000, −10.80807158993283, −10.36458474398657, −10.03239958908395, −9.761068698884786, −9.045389067931086, −8.536805306318145, −8.235794355989974, −7.605660952066497, −7.426161567624092, −6.533607681095132, −6.263507083580006, −5.943438741201587, −5.404781230329834, −4.747693541488167, −4.211007677004945, −3.708655844607091, −3.011419848725195, −2.585337317995472, −1.835832352564830, −1.344630146404482, −0.5911756636439939, 0,
0.5911756636439939, 1.344630146404482, 1.835832352564830, 2.585337317995472, 3.011419848725195, 3.708655844607091, 4.211007677004945, 4.747693541488167, 5.404781230329834, 5.943438741201587, 6.263507083580006, 6.533607681095132, 7.426161567624092, 7.605660952066497, 8.235794355989974, 8.536805306318145, 9.045389067931086, 9.761068698884786, 10.03239958908395, 10.36458474398657, 10.80807158993283, 11.37227796139000, 11.76704423518807, 11.99257100011101, 12.81502649820538