| L(s) = 1 | + 3-s − 2.13·7-s + 9-s − 3.02·11-s + 13-s − 1.62·17-s + 7.08·19-s − 2.13·21-s + 3.62·23-s + 27-s − 2.46·29-s − 8.37·31-s − 3.02·33-s + 2.65·37-s + 39-s + 4.19·41-s + 4.76·43-s − 7.11·47-s − 2.46·49-s − 1.62·51-s + 7.58·53-s + 7.08·57-s + 14.6·59-s − 14.0·61-s − 2.13·63-s + 1.01·67-s + 3.62·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.805·7-s + 0.333·9-s − 0.912·11-s + 0.277·13-s − 0.393·17-s + 1.62·19-s − 0.464·21-s + 0.755·23-s + 0.192·27-s − 0.457·29-s − 1.50·31-s − 0.526·33-s + 0.436·37-s + 0.160·39-s + 0.655·41-s + 0.726·43-s − 1.03·47-s − 0.351·49-s − 0.227·51-s + 1.04·53-s + 0.938·57-s + 1.90·59-s − 1.79·61-s − 0.268·63-s + 0.123·67-s + 0.436·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.055888987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.055888987\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2.13T + 7T^{2} \) |
| 11 | \( 1 + 3.02T + 11T^{2} \) |
| 17 | \( 1 + 1.62T + 17T^{2} \) |
| 19 | \( 1 - 7.08T + 19T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 + 8.37T + 31T^{2} \) |
| 37 | \( 1 - 2.65T + 37T^{2} \) |
| 41 | \( 1 - 4.19T + 41T^{2} \) |
| 43 | \( 1 - 4.76T + 43T^{2} \) |
| 47 | \( 1 + 7.11T + 47T^{2} \) |
| 53 | \( 1 - 7.58T + 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 1.01T + 67T^{2} \) |
| 71 | \( 1 - 9.32T + 71T^{2} \) |
| 73 | \( 1 - 2.34T + 73T^{2} \) |
| 79 | \( 1 + 7.87T + 79T^{2} \) |
| 83 | \( 1 - 0.396T + 83T^{2} \) |
| 89 | \( 1 - 6.52T + 89T^{2} \) |
| 97 | \( 1 + 0.794T + 97T^{2} \) |
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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
−7.72297094938744737221373578519, −7.31372395106922088722067785137, −6.59636145473823802041486503893, −5.66179686459640340967112029825, −5.17549359045505895476272952567, −4.15719934003535503809530744661, −3.34178717156770613489131993537, −2.85877761733018880994193892277, −1.90097748573005664856536390881, −0.68131317992761279506823685965,
0.68131317992761279506823685965, 1.90097748573005664856536390881, 2.85877761733018880994193892277, 3.34178717156770613489131993537, 4.15719934003535503809530744661, 5.17549359045505895476272952567, 5.66179686459640340967112029825, 6.59636145473823802041486503893, 7.31372395106922088722067785137, 7.72297094938744737221373578519
