L(s) = 1 | + 0.414·3-s − 2.82·9-s − 11-s − 2.41·13-s + 0.414·17-s + 2·19-s + 2.24·23-s − 2.41·27-s + 29-s + 1.75·31-s − 0.414·33-s + 7.89·37-s − 0.999·39-s + 7.41·41-s + 0.343·43-s − 10.4·47-s + 0.171·51-s − 9.41·53-s + 0.828·57-s + 10.2·59-s − 1.17·61-s − 1.41·67-s + 0.928·69-s + 14.4·71-s + 5.17·73-s − 14.6·79-s + 7.48·81-s + ⋯ |
L(s) = 1 | + 0.239·3-s − 0.942·9-s − 0.301·11-s − 0.669·13-s + 0.100·17-s + 0.458·19-s + 0.467·23-s − 0.464·27-s + 0.185·29-s + 0.315·31-s − 0.0721·33-s + 1.29·37-s − 0.160·39-s + 1.15·41-s + 0.0523·43-s − 1.51·47-s + 0.0240·51-s − 1.29·53-s + 0.109·57-s + 1.33·59-s − 0.150·61-s − 0.172·67-s + 0.111·69-s + 1.71·71-s + 0.605·73-s − 1.64·79-s + 0.831·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 17 | \( 1 - 0.414T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 2.24T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 - 7.89T + 37T^{2} \) |
| 41 | \( 1 - 7.41T + 41T^{2} \) |
| 43 | \( 1 - 0.343T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 9.41T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 1.17T + 61T^{2} \) |
| 67 | \( 1 + 1.41T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 1.65T + 89T^{2} \) |
| 97 | \( 1 - 0.0710T + 97T^{2} \) |
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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
−7.42735595810128330478000457348, −6.65692833283134968216736881342, −5.93790018229497773635797935430, −5.26640716718316782132296590146, −4.64802319246514863998535508912, −3.72332938339235997388163599074, −2.85562725795060132773501730464, −2.43593051239561664514041145368, −1.18490698918217620593356052274, 0,
1.18490698918217620593356052274, 2.43593051239561664514041145368, 2.85562725795060132773501730464, 3.72332938339235997388163599074, 4.64802319246514863998535508912, 5.26640716718316782132296590146, 5.93790018229497773635797935430, 6.65692833283134968216736881342, 7.42735595810128330478000457348