| L(s) = 1 | + 1.83·2-s + 1.37·4-s + 3.54·7-s − 1.14·8-s + 1.74·11-s − 1.99·13-s + 6.51·14-s − 4.85·16-s + 6.52·17-s − 4.61·19-s + 3.19·22-s − 9.26·23-s − 3.66·26-s + 4.87·28-s + 9.40·29-s + 7.06·31-s − 6.63·32-s + 11.9·34-s − 37-s − 8.48·38-s + 2.07·41-s + 5.12·43-s + 2.39·44-s − 17.0·46-s + 8.12·47-s + 5.58·49-s − 2.73·52-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.687·4-s + 1.34·7-s − 0.406·8-s + 0.525·11-s − 0.552·13-s + 1.74·14-s − 1.21·16-s + 1.58·17-s − 1.05·19-s + 0.682·22-s − 1.93·23-s − 0.717·26-s + 0.921·28-s + 1.74·29-s + 1.26·31-s − 1.17·32-s + 2.05·34-s − 0.164·37-s − 1.37·38-s + 0.323·41-s + 0.780·43-s + 0.361·44-s − 2.51·46-s + 1.18·47-s + 0.798·49-s − 0.379·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.802902165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.802902165\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 1.83T + 2T^{2} \) |
| 7 | \( 1 - 3.54T + 7T^{2} \) |
| 11 | \( 1 - 1.74T + 11T^{2} \) |
| 13 | \( 1 + 1.99T + 13T^{2} \) |
| 17 | \( 1 - 6.52T + 17T^{2} \) |
| 19 | \( 1 + 4.61T + 19T^{2} \) |
| 23 | \( 1 + 9.26T + 23T^{2} \) |
| 29 | \( 1 - 9.40T + 29T^{2} \) |
| 31 | \( 1 - 7.06T + 31T^{2} \) |
| 41 | \( 1 - 2.07T + 41T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 - 8.12T + 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 - 1.14T + 59T^{2} \) |
| 61 | \( 1 + 2.40T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 - 5.13T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 7.83T + 79T^{2} \) |
| 83 | \( 1 + 5.45T + 83T^{2} \) |
| 89 | \( 1 + 4.14T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.77027432964681639890922088122, −6.97936487917321086352029542771, −6.05659200553274862061392653175, −5.72428849522135175290106485071, −4.81632024504647825363983329791, −4.36475267463626035089634346725, −3.80595571746826355500786610401, −2.73558937701165853347356407623, −2.06712875927414212374639836244, −0.906800132124923830822231339161,
0.906800132124923830822231339161, 2.06712875927414212374639836244, 2.73558937701165853347356407623, 3.80595571746826355500786610401, 4.36475267463626035089634346725, 4.81632024504647825363983329791, 5.72428849522135175290106485071, 6.05659200553274862061392653175, 6.97936487917321086352029542771, 7.77027432964681639890922088122
