| L(s) = 1 | + 3-s + 4·7-s − 2·9-s + 11-s − 6·13-s + 3·17-s + 19-s + 4·21-s − 6·23-s − 5·25-s − 5·27-s + 4·29-s + 9·31-s + 33-s + 10·37-s − 6·39-s + 10·41-s + 5·43-s + 8·47-s + 9·49-s + 3·51-s + 53-s + 57-s + 12·59-s + 12·61-s − 8·63-s − 67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s − 2/3·9-s + 0.301·11-s − 1.66·13-s + 0.727·17-s + 0.229·19-s + 0.872·21-s − 1.25·23-s − 25-s − 0.962·27-s + 0.742·29-s + 1.61·31-s + 0.174·33-s + 1.64·37-s − 0.960·39-s + 1.56·41-s + 0.762·43-s + 1.16·47-s + 9/7·49-s + 0.420·51-s + 0.137·53-s + 0.132·57-s + 1.56·59-s + 1.53·61-s − 1.00·63-s − 0.122·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.591149665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.591149665\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−8.202496511471742924968268571352, −7.86572285846872318637757801828, −7.34051177920821357857404532617, −6.06989100207008772720957786126, −5.46922148139898728624776399447, −4.56972457413870963127025122357, −3.99967031081932524625426848672, −2.61332450577590056550735256328, −2.26683558914784140352553351249, −0.908639425259354542582191970542,
0.908639425259354542582191970542, 2.26683558914784140352553351249, 2.61332450577590056550735256328, 3.99967031081932524625426848672, 4.56972457413870963127025122357, 5.46922148139898728624776399447, 6.06989100207008772720957786126, 7.34051177920821357857404532617, 7.86572285846872318637757801828, 8.202496511471742924968268571352
