| L(s) = 1 | − 3-s + 5-s + 9-s − 13-s − 15-s + 2·17-s + 3·19-s + 6·23-s + 25-s − 27-s − 6·29-s + 7·31-s + 37-s + 39-s + 4·41-s − 11·43-s + 45-s + 6·47-s − 2·51-s − 4·53-s − 3·57-s − 14·59-s − 6·61-s − 65-s + 3·67-s − 6·69-s + 7·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.277·13-s − 0.258·15-s + 0.485·17-s + 0.688·19-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.25·31-s + 0.164·37-s + 0.160·39-s + 0.624·41-s − 1.67·43-s + 0.149·45-s + 0.875·47-s − 0.280·51-s − 0.549·53-s − 0.397·57-s − 1.82·59-s − 0.768·61-s − 0.124·65-s + 0.366·67-s − 0.722·69-s + 0.819·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.799426494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.799426494\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.912313759708136519136820802319, −7.40300787743763199879857778368, −6.56211341475152102234005813412, −6.00219824109562165146049783401, −5.14448130416756154606792542279, −4.75333864196645054733192775317, −3.60120941711706682351116205749, −2.82549383595952350719926255169, −1.72340021257728204028496101118, −0.75471966280999171908094685909,
0.75471966280999171908094685909, 1.72340021257728204028496101118, 2.82549383595952350719926255169, 3.60120941711706682351116205749, 4.75333864196645054733192775317, 5.14448130416756154606792542279, 6.00219824109562165146049783401, 6.56211341475152102234005813412, 7.40300787743763199879857778368, 7.912313759708136519136820802319
