| L(s) = 1 | + 1.61·2-s + 0.618·4-s + 0.671·7-s − 2.23·8-s + 3.14·11-s + 5.09·13-s + 1.08·14-s − 4.85·16-s + 5.91·17-s − 0.179·19-s + 5.09·22-s − 4.89·23-s + 8.24·26-s + 0.415·28-s + 0.430·29-s − 1.70·31-s − 3.38·32-s + 9.57·34-s + 37-s − 0.289·38-s + 11.0·41-s − 3.00·43-s + 1.94·44-s − 7.91·46-s − 6.48·47-s − 6.54·49-s + 3.14·52-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.309·4-s + 0.253·7-s − 0.790·8-s + 0.949·11-s + 1.41·13-s + 0.290·14-s − 1.21·16-s + 1.43·17-s − 0.0411·19-s + 1.08·22-s − 1.02·23-s + 1.61·26-s + 0.0784·28-s + 0.0800·29-s − 0.306·31-s − 0.597·32-s + 1.64·34-s + 0.164·37-s − 0.0470·38-s + 1.73·41-s − 0.458·43-s + 0.293·44-s − 1.16·46-s − 0.945·47-s − 0.935·49-s + 0.436·52-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
| \(L(1)\) |
\(\approx\) |
\(4.121716248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.121716248\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.61T + 2T^{2} \) |
| 7 | \( 1 - 0.671T + 7T^{2} \) |
| 11 | \( 1 - 3.14T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 - 5.91T + 17T^{2} \) |
| 19 | \( 1 + 0.179T + 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 - 0.430T + 29T^{2} \) |
| 31 | \( 1 + 1.70T + 31T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 3.00T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 + 1.64T + 53T^{2} \) |
| 59 | \( 1 - 3.40T + 59T^{2} \) |
| 61 | \( 1 - 4.00T + 61T^{2} \) |
| 67 | \( 1 - 8.44T + 67T^{2} \) |
| 71 | \( 1 - 4.41T + 71T^{2} \) |
| 73 | \( 1 - 5.40T + 73T^{2} \) |
| 79 | \( 1 - 9.01T + 79T^{2} \) |
| 83 | \( 1 - 5.82T + 83T^{2} \) |
| 89 | \( 1 + 2.45T + 89T^{2} \) |
| 97 | \( 1 - 5.56T + 97T^{2} \) |
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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
−7.937454191019159502004087607034, −6.75159234350215285960863493849, −6.26310848121439087983453889571, −5.66527092977901335470931627689, −5.04260048939275797184889873049, −4.04883461465850380692564785841, −3.76561096474572914710199790302, −3.02115317968378008075435973809, −1.85564867522888367147581607872, −0.879306183334686432508329183767,
0.879306183334686432508329183767, 1.85564867522888367147581607872, 3.02115317968378008075435973809, 3.76561096474572914710199790302, 4.04883461465850380692564785841, 5.04260048939275797184889873049, 5.66527092977901335470931627689, 6.26310848121439087983453889571, 6.75159234350215285960863493849, 7.937454191019159502004087607034
