| L(s) = 1 | − 3·3-s − 10·7-s + 9·9-s − 46·11-s + 34·13-s − 66·17-s + 104·19-s + 30·21-s − 164·23-s − 27·27-s + 224·29-s − 72·31-s + 138·33-s + 22·37-s − 102·39-s + 194·41-s − 108·43-s + 480·47-s − 243·49-s + 198·51-s − 286·53-s − 312·57-s + 426·59-s + 698·61-s − 90·63-s − 328·67-s + 492·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.539·7-s + 1/3·9-s − 1.26·11-s + 0.725·13-s − 0.941·17-s + 1.25·19-s + 0.311·21-s − 1.48·23-s − 0.192·27-s + 1.43·29-s − 0.417·31-s + 0.727·33-s + 0.0977·37-s − 0.418·39-s + 0.738·41-s − 0.383·43-s + 1.48·47-s − 0.708·49-s + 0.543·51-s − 0.741·53-s − 0.725·57-s + 0.940·59-s + 1.46·61-s − 0.179·63-s − 0.598·67-s + 0.858·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.159822668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.159822668\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 10 T + p^{3} T^{2} \) |
| 11 | \( 1 + 46 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 104 T + p^{3} T^{2} \) |
| 23 | \( 1 + 164 T + p^{3} T^{2} \) |
| 29 | \( 1 - 224 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 - 22 T + p^{3} T^{2} \) |
| 41 | \( 1 - 194 T + p^{3} T^{2} \) |
| 43 | \( 1 + 108 T + p^{3} T^{2} \) |
| 47 | \( 1 - 480 T + p^{3} T^{2} \) |
| 53 | \( 1 + 286 T + p^{3} T^{2} \) |
| 59 | \( 1 - 426 T + p^{3} T^{2} \) |
| 61 | \( 1 - 698 T + p^{3} T^{2} \) |
| 67 | \( 1 + 328 T + p^{3} T^{2} \) |
| 71 | \( 1 - 188 T + p^{3} T^{2} \) |
| 73 | \( 1 - 740 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1168 T + p^{3} T^{2} \) |
| 83 | \( 1 + 412 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1206 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1384 T + p^{3} T^{2} \) |
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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
−10.31361400282571948570298361642, −9.577545049411305744691084873516, −8.442444840359026104247417519191, −7.57936441709628640839700953356, −6.53110891973689591665966778579, −5.74355868377153774783197715275, −4.78416856826044504590706125789, −3.58134630986534094951162009050, −2.31621239463665470247720721182, −0.64387158494190331908570970942,
0.64387158494190331908570970942, 2.31621239463665470247720721182, 3.58134630986534094951162009050, 4.78416856826044504590706125789, 5.74355868377153774783197715275, 6.53110891973689591665966778579, 7.57936441709628640839700953356, 8.442444840359026104247417519191, 9.577545049411305744691084873516, 10.31361400282571948570298361642
