| L(s) = 1 | − 2.59·2-s + 4.74·4-s + 3.28·7-s − 7.12·8-s − 4.30·11-s − 3.46·13-s − 8.52·14-s + 9.02·16-s + 5.44·17-s − 7.63·19-s + 11.1·22-s + 5.04·23-s + 8.98·26-s + 15.5·28-s + 3.12·29-s − 2.06·31-s − 9.18·32-s − 14.1·34-s − 1.89·37-s + 19.8·38-s − 3.89·41-s − 3.20·43-s − 20.4·44-s − 13.1·46-s − 6.28·47-s + 3.78·49-s − 16.4·52-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 2.37·4-s + 1.24·7-s − 2.52·8-s − 1.29·11-s − 0.959·13-s − 2.27·14-s + 2.25·16-s + 1.32·17-s − 1.75·19-s + 2.38·22-s + 1.05·23-s + 1.76·26-s + 2.94·28-s + 0.579·29-s − 0.371·31-s − 1.62·32-s − 2.42·34-s − 0.310·37-s + 3.21·38-s − 0.608·41-s − 0.489·43-s − 3.08·44-s − 1.93·46-s − 0.916·47-s + 0.540·49-s − 2.27·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6764422472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6764422472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 7 | \( 1 - 3.28T + 7T^{2} \) |
| 11 | \( 1 + 4.30T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 5.44T + 17T^{2} \) |
| 19 | \( 1 + 7.63T + 19T^{2} \) |
| 23 | \( 1 - 5.04T + 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 + 1.89T + 37T^{2} \) |
| 41 | \( 1 + 3.89T + 41T^{2} \) |
| 43 | \( 1 + 3.20T + 43T^{2} \) |
| 47 | \( 1 + 6.28T + 47T^{2} \) |
| 53 | \( 1 - 2.51T + 53T^{2} \) |
| 59 | \( 1 + 7.72T + 59T^{2} \) |
| 61 | \( 1 + 2.95T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 4.72T + 71T^{2} \) |
| 73 | \( 1 + 1.64T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 5.01T + 83T^{2} \) |
| 89 | \( 1 - 9.00T + 89T^{2} \) |
| 97 | \( 1 - 2.57T + 97T^{2} \) |
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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
−8.191051328503361635832536328949, −7.72905422317064543479091097009, −7.11061441481255038345960955071, −6.31795156933692083178459706247, −5.24929580987139655519641175112, −4.76655149340973200711708992045, −3.24014217650319813816022431373, −2.33124120240113891074944296559, −1.72832986034587190468033135267, −0.57104155544024247155956008011,
0.57104155544024247155956008011, 1.72832986034587190468033135267, 2.33124120240113891074944296559, 3.24014217650319813816022431373, 4.76655149340973200711708992045, 5.24929580987139655519641175112, 6.31795156933692083178459706247, 7.11061441481255038345960955071, 7.72905422317064543479091097009, 8.191051328503361635832536328949
