| L(s) = 1 | − 1.13·2-s − 0.714·4-s − 2.46·7-s + 3.07·8-s − 1.71·11-s − 6.49·13-s + 2.79·14-s − 2.05·16-s + 3.32·17-s + 0.734·19-s + 1.94·22-s − 2.08·23-s + 7.35·26-s + 1.75·28-s + 4.21·29-s + 7.46·31-s − 3.82·32-s − 3.77·34-s − 37-s − 0.832·38-s − 1.71·41-s − 1.81·43-s + 1.22·44-s + 2.36·46-s − 0.882·47-s − 0.940·49-s + 4.64·52-s + ⋯ |
| L(s) = 1 | − 0.801·2-s − 0.357·4-s − 0.930·7-s + 1.08·8-s − 0.517·11-s − 1.80·13-s + 0.745·14-s − 0.514·16-s + 0.807·17-s + 0.168·19-s + 0.414·22-s − 0.434·23-s + 1.44·26-s + 0.332·28-s + 0.782·29-s + 1.34·31-s − 0.675·32-s − 0.647·34-s − 0.164·37-s − 0.135·38-s − 0.267·41-s − 0.277·43-s + 0.184·44-s + 0.348·46-s − 0.128·47-s − 0.134·49-s + 0.643·52-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
| \(L(1)\) |
\(\approx\) |
\(0.3971718618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3971718618\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.13T + 2T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 + 1.71T + 11T^{2} \) |
| 13 | \( 1 + 6.49T + 13T^{2} \) |
| 17 | \( 1 - 3.32T + 17T^{2} \) |
| 19 | \( 1 - 0.734T + 19T^{2} \) |
| 23 | \( 1 + 2.08T + 23T^{2} \) |
| 29 | \( 1 - 4.21T + 29T^{2} \) |
| 31 | \( 1 - 7.46T + 31T^{2} \) |
| 41 | \( 1 + 1.71T + 41T^{2} \) |
| 43 | \( 1 + 1.81T + 43T^{2} \) |
| 47 | \( 1 + 0.882T + 47T^{2} \) |
| 53 | \( 1 + 7.03T + 53T^{2} \) |
| 59 | \( 1 + 0.387T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 - 8.23T + 79T^{2} \) |
| 83 | \( 1 + 4.80T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−7.67837610917937041291412125281, −7.47700188613210667012071918688, −6.54999698995203528559381657090, −5.78997079400822158350809652914, −4.82722179423480275318595960959, −4.52096752941316051191035301005, −3.26723971864630802435965624805, −2.70165082283384309180285601532, −1.55246960971510616373375152274, −0.35409085468885165697587350055,
0.35409085468885165697587350055, 1.55246960971510616373375152274, 2.70165082283384309180285601532, 3.26723971864630802435965624805, 4.52096752941316051191035301005, 4.82722179423480275318595960959, 5.78997079400822158350809652914, 6.54999698995203528559381657090, 7.47700188613210667012071918688, 7.67837610917937041291412125281
