| L(s) = 1 | + 2.19·3-s − 0.635·5-s + 1.83·9-s + 11-s − 1.80·13-s − 1.39·15-s − 2.83·17-s + 5.56·19-s − 2.16·23-s − 4.59·25-s − 2.56·27-s − 10.4·29-s − 6.43·31-s + 2.19·33-s + 6.06·37-s − 3.96·39-s + 7.53·41-s + 4.86·43-s − 1.16·45-s + 2.83·47-s − 6.23·51-s + 7.46·53-s − 0.635·55-s + 12.2·57-s − 11.8·59-s − 4.33·61-s + 1.14·65-s + ⋯ |
| L(s) = 1 | + 1.26·3-s − 0.284·5-s + 0.611·9-s + 0.301·11-s − 0.499·13-s − 0.360·15-s − 0.687·17-s + 1.27·19-s − 0.451·23-s − 0.919·25-s − 0.493·27-s − 1.93·29-s − 1.15·31-s + 0.382·33-s + 0.997·37-s − 0.634·39-s + 1.17·41-s + 0.742·43-s − 0.173·45-s + 0.413·47-s − 0.872·51-s + 1.02·53-s − 0.0856·55-s + 1.62·57-s − 1.53·59-s − 0.554·61-s + 0.141·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2.19T + 3T^{2} \) |
| 5 | \( 1 + 0.635T + 5T^{2} \) |
| 13 | \( 1 + 1.80T + 13T^{2} \) |
| 17 | \( 1 + 2.83T + 17T^{2} \) |
| 19 | \( 1 - 5.56T + 19T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 6.43T + 31T^{2} \) |
| 37 | \( 1 - 6.06T + 37T^{2} \) |
| 41 | \( 1 - 7.53T + 41T^{2} \) |
| 43 | \( 1 - 4.86T + 43T^{2} \) |
| 47 | \( 1 - 2.83T + 47T^{2} \) |
| 53 | \( 1 - 7.46T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 4.33T + 61T^{2} \) |
| 67 | \( 1 - 1.60T + 67T^{2} \) |
| 71 | \( 1 + 4.29T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 + 4.76T + 79T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 - 0.364T + 89T^{2} \) |
| 97 | \( 1 + 2.59T + 97T^{2} \) |
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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
−7.63491550300666678025156836506, −7.10364298926763192359338759541, −5.94065513104705006758348180885, −5.46263130391021075909639756851, −4.23316110447534602139615908279, −3.90241172208552138177481575831, −3.01748196908401718506560306975, −2.33302853413299642984041036805, −1.51471020276721031040875107841, 0,
1.51471020276721031040875107841, 2.33302853413299642984041036805, 3.01748196908401718506560306975, 3.90241172208552138177481575831, 4.23316110447534602139615908279, 5.46263130391021075909639756851, 5.94065513104705006758348180885, 7.10364298926763192359338759541, 7.63491550300666678025156836506
