| L(s) = 1 | + 4.68·3-s + 12.9·9-s + 11.4·11-s + 6.18·13-s + 25·25-s + 18.4·27-s − 14.8·29-s + 38.7·31-s + 53.7·33-s + 28.9·39-s − 84.2·43-s + 49·49-s − 29.3·53-s + 40.7·59-s − 106.·61-s + 124.·67-s + 141.·71-s + 145.·73-s + 117.·75-s − 29.9·81-s − 158.·83-s − 69.4·87-s + 132.·89-s + 181.·93-s + 148.·99-s − 78.6·107-s − 25.3·109-s + ⋯ |
| L(s) = 1 | + 1.56·3-s + 1.43·9-s + 1.04·11-s + 0.476·13-s + 25-s + 0.683·27-s − 0.511·29-s + 1.24·31-s + 1.62·33-s + 0.743·39-s − 1.95·43-s + 0.999·49-s − 0.554·53-s + 0.690·59-s − 1.74·61-s + 1.86·67-s + 1.99·71-s + 1.99·73-s + 1.56·75-s − 0.370·81-s − 1.91·83-s − 0.798·87-s + 1.49·89-s + 1.94·93-s + 1.50·99-s − 0.735·107-s − 0.232·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1388 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1388 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.990975034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.990975034\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 347 | \( 1 + 347T \) |
| good | 3 | \( 1 - 4.68T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 - 11.4T + 121T^{2} \) |
| 13 | \( 1 - 6.18T + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 14.8T + 841T^{2} \) |
| 31 | \( 1 - 38.7T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 84.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 29.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 40.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 106.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 124.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 141.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 145.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 158.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 132.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.291011654450904109860903581266, −8.530681858776366046646523494202, −8.073595207341210340936544480714, −7.01150921253656839309527242979, −6.36683616336466128378278165569, −4.99148044251890458958235568381, −3.93367891521433850182160026546, −3.30365927965141300516687246006, −2.26933973759486599997344620197, −1.18409616412781294225059480913,
1.18409616412781294225059480913, 2.26933973759486599997344620197, 3.30365927965141300516687246006, 3.93367891521433850182160026546, 4.99148044251890458958235568381, 6.36683616336466128378278165569, 7.01150921253656839309527242979, 8.073595207341210340936544480714, 8.530681858776366046646523494202, 9.291011654450904109860903581266
