| L(s) = 1 | + 0.386·3-s − 3.17·5-s + 3.44·7-s − 2.85·9-s + 3.44·11-s − 1.22·15-s − 1.77·17-s + 3.33·19-s + 1.33·21-s − 0.386·23-s + 5.07·25-s − 2.26·27-s − 4.57·29-s + 11.0·31-s + 1.33·33-s − 10.9·35-s + 1.62·37-s − 2.26·41-s + 10.7·43-s + 9.04·45-s + 8.11·47-s + 4.85·49-s − 0.685·51-s + 11.6·53-s − 10.9·55-s + 1.28·57-s − 6.32·59-s + ⋯ |
| L(s) = 1 | + 0.223·3-s − 1.41·5-s + 1.30·7-s − 0.950·9-s + 1.03·11-s − 0.316·15-s − 0.430·17-s + 0.764·19-s + 0.290·21-s − 0.0805·23-s + 1.01·25-s − 0.435·27-s − 0.849·29-s + 1.98·31-s + 0.231·33-s − 1.84·35-s + 0.266·37-s − 0.354·41-s + 1.63·43-s + 1.34·45-s + 1.18·47-s + 0.692·49-s − 0.0959·51-s + 1.60·53-s − 1.47·55-s + 0.170·57-s − 0.823·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.553886952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.553886952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.386T + 3T^{2} \) |
| 5 | \( 1 + 3.17T + 5T^{2} \) |
| 7 | \( 1 - 3.44T + 7T^{2} \) |
| 11 | \( 1 - 3.44T + 11T^{2} \) |
| 17 | \( 1 + 1.77T + 17T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 + 0.386T + 23T^{2} \) |
| 29 | \( 1 + 4.57T + 29T^{2} \) |
| 31 | \( 1 - 11.0T + 31T^{2} \) |
| 37 | \( 1 - 1.62T + 37T^{2} \) |
| 41 | \( 1 + 2.26T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 8.11T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 6.32T + 59T^{2} \) |
| 61 | \( 1 - 2.42T + 61T^{2} \) |
| 67 | \( 1 + 9.14T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 8.40T + 73T^{2} \) |
| 79 | \( 1 - 8.22T + 79T^{2} \) |
| 83 | \( 1 - 1.11T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 + 4.48T + 97T^{2} \) |
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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
−9.312367171748487409818235218862, −8.679968160619429986099194666623, −7.952642809399912858354916532150, −7.49235489089314377390541740681, −6.37832649731518325052002481767, −5.27166226261847343622726576189, −4.34227035169108483994197024365, −3.68358125279237940287840289395, −2.46119928382055693083293082279, −0.928047109415552117017032761887,
0.928047109415552117017032761887, 2.46119928382055693083293082279, 3.68358125279237940287840289395, 4.34227035169108483994197024365, 5.27166226261847343622726576189, 6.37832649731518325052002481767, 7.49235489089314377390541740681, 7.952642809399912858354916532150, 8.679968160619429986099194666623, 9.312367171748487409818235218862
