L(s) = 1 | − 4·5-s − 4·11-s + 4·13-s − 4·19-s + 11·25-s − 2·29-s − 8·31-s − 6·37-s − 4·43-s − 8·47-s + 10·53-s + 16·55-s + 4·59-s − 4·61-s − 16·65-s − 4·67-s + 8·71-s − 16·73-s + 8·79-s − 12·83-s − 8·89-s + 16·95-s + 8·97-s − 4·101-s + 8·103-s + 4·107-s − 14·109-s + ⋯ |
L(s) = 1 | − 1.78·5-s − 1.20·11-s + 1.10·13-s − 0.917·19-s + 11/5·25-s − 0.371·29-s − 1.43·31-s − 0.986·37-s − 0.609·43-s − 1.16·47-s + 1.37·53-s + 2.15·55-s + 0.520·59-s − 0.512·61-s − 1.98·65-s − 0.488·67-s + 0.949·71-s − 1.87·73-s + 0.900·79-s − 1.31·83-s − 0.847·89-s + 1.64·95-s + 0.812·97-s − 0.398·101-s + 0.788·103-s + 0.386·107-s − 1.34·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6096210379\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6096210379\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.970538464910318214187820369382, −7.33635177401815799004770129618, −6.76164809946628399492885993210, −5.76665958552437262230380818520, −5.03044683603915170224456791587, −4.20845716289353478865350204608, −3.63247435610844950456488595214, −2.97184008259167031693015334029, −1.76661944201507191074414099062, −0.38676708206125813281894201513,
0.38676708206125813281894201513, 1.76661944201507191074414099062, 2.97184008259167031693015334029, 3.63247435610844950456488595214, 4.20845716289353478865350204608, 5.03044683603915170224456791587, 5.76665958552437262230380818520, 6.76164809946628399492885993210, 7.33635177401815799004770129618, 7.970538464910318214187820369382