| L(s) = 1 | + 3-s − 5-s − 3·7-s − 2·9-s − 2·11-s − 15-s − 3·17-s − 2·19-s − 3·21-s + 4·23-s − 4·25-s − 5·27-s + 2·29-s − 4·31-s − 2·33-s + 3·35-s − 5·37-s + 12·41-s + 7·43-s + 2·45-s + 9·47-s + 2·49-s − 3·51-s + 4·53-s + 2·55-s − 2·57-s − 6·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.13·7-s − 2/3·9-s − 0.603·11-s − 0.258·15-s − 0.727·17-s − 0.458·19-s − 0.654·21-s + 0.834·23-s − 4/5·25-s − 0.962·27-s + 0.371·29-s − 0.718·31-s − 0.348·33-s + 0.507·35-s − 0.821·37-s + 1.87·41-s + 1.06·43-s + 0.298·45-s + 1.31·47-s + 2/7·49-s − 0.420·51-s + 0.549·53-s + 0.269·55-s − 0.264·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.057888433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.057888433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.155475919227474927845697076584, −7.53790637677321817500988270720, −6.82225463998717423647601419811, −6.02768784847994320314313833225, −5.38702250040064766465923905587, −4.27972725773597363535034100520, −3.61924269731192216193712406886, −2.81310616067439690500847810982, −2.22011891993564117903631356829, −0.50089057585371175170833868276,
0.50089057585371175170833868276, 2.22011891993564117903631356829, 2.81310616067439690500847810982, 3.61924269731192216193712406886, 4.27972725773597363535034100520, 5.38702250040064766465923905587, 6.02768784847994320314313833225, 6.82225463998717423647601419811, 7.53790637677321817500988270720, 8.155475919227474927845697076584
