| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 4·7-s + 8-s + 9-s + 10-s − 12-s − 13-s − 4·14-s − 15-s + 16-s − 6·17-s + 18-s + 8·19-s + 20-s + 4·21-s − 6·23-s − 24-s + 25-s − 26-s − 27-s − 4·28-s + 6·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.872·21-s − 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.755·28-s + 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.268090439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.268090439\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.35156615385571, −12.16179281815374, −11.67779265839669, −11.37155046499576, −10.52345097660515, −10.26188637933318, −9.951674639192481, −9.462225341108527, −8.903928151097714, −8.524391486064236, −7.610790167922368, −7.309394667399062, −6.728436511246796, −6.425748932882754, −6.028558509248787, −5.548204182879084, −4.995763379058166, −4.610514883794238, −3.925849879313497, −3.427077560841597, −3.025280133521564, −2.328484389295885, −1.929272653790604, −1.008885290598213, −0.3882738271288198,
0.3882738271288198, 1.008885290598213, 1.929272653790604, 2.328484389295885, 3.025280133521564, 3.427077560841597, 3.925849879313497, 4.610514883794238, 4.995763379058166, 5.548204182879084, 6.028558509248787, 6.425748932882754, 6.728436511246796, 7.309394667399062, 7.610790167922368, 8.524391486064236, 8.903928151097714, 9.462225341108527, 9.951674639192481, 10.26188637933318, 10.52345097660515, 11.37155046499576, 11.67779265839669, 12.16179281815374, 12.35156615385571
