| L(s) = 1 | − 0.209·2-s + 1.33·3-s − 0.956·4-s − 0.279·6-s + 1.82·7-s + 0.408·8-s + 0.790·9-s − 1.27·12-s − 0.381·14-s + 0.870·16-s − 1.95·17-s − 0.165·18-s + 2.44·21-s + 0.547·24-s − 0.279·27-s − 1.74·28-s − 0.591·32-s + 0.408·34-s − 0.756·36-s + 0.618·37-s − 0.511·42-s + 47-s + 1.16·48-s + 2.33·49-s − 2.61·51-s − 1.61·53-s + 0.0584·54-s + ⋯ |
| L(s) = 1 | − 0.209·2-s + 1.33·3-s − 0.956·4-s − 0.279·6-s + 1.82·7-s + 0.408·8-s + 0.790·9-s − 1.27·12-s − 0.381·14-s + 0.870·16-s − 1.95·17-s − 0.165·18-s + 2.44·21-s + 0.547·24-s − 0.279·27-s − 1.74·28-s − 0.591·32-s + 0.408·34-s − 0.756·36-s + 0.618·37-s − 0.511·42-s + 47-s + 1.16·48-s + 2.33·49-s − 2.61·51-s − 1.61·53-s + 0.0584·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.361594636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.361594636\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 0.209T + T^{2} \) |
| 3 | \( 1 - 1.33T + T^{2} \) |
| 7 | \( 1 - 1.82T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.95T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.618T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.61T + T^{2} \) |
| 59 | \( 1 - 1.82T + T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.33T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 0.618T + T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 + 1.61T + T^{2} \) |
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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
−9.583348054982151544546787868698, −9.006430318919190363572171260429, −8.289595843895530691205373993781, −8.032605322128588500373370554111, −7.01186416378214759112050198646, −5.49362399381875643287340414447, −4.52100531931816140155794660870, −4.05421163363881325904602017956, −2.58721274056214944339234470054, −1.60328329860327961264197432981,
1.60328329860327961264197432981, 2.58721274056214944339234470054, 4.05421163363881325904602017956, 4.52100531931816140155794660870, 5.49362399381875643287340414447, 7.01186416378214759112050198646, 8.032605322128588500373370554111, 8.289595843895530691205373993781, 9.006430318919190363572171260429, 9.583348054982151544546787868698
