| L(s) = 1 | − 3·3-s − 10·5-s − 8·7-s + 9·9-s − 40·11-s − 13·13-s + 30·15-s + 130·17-s + 20·19-s + 24·21-s − 25·25-s − 27·27-s + 18·29-s − 184·31-s + 120·33-s + 80·35-s + 74·37-s + 39·39-s − 362·41-s − 76·43-s − 90·45-s − 452·47-s − 279·49-s − 390·51-s − 382·53-s + 400·55-s − 60·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.431·7-s + 1/3·9-s − 1.09·11-s − 0.277·13-s + 0.516·15-s + 1.85·17-s + 0.241·19-s + 0.249·21-s − 1/5·25-s − 0.192·27-s + 0.115·29-s − 1.06·31-s + 0.633·33-s + 0.386·35-s + 0.328·37-s + 0.160·39-s − 1.37·41-s − 0.269·43-s − 0.298·45-s − 1.40·47-s − 0.813·49-s − 1.07·51-s − 0.990·53-s + 0.980·55-s − 0.139·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5369921535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5369921535\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 17 | \( 1 - 130 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 18 T + p^{3} T^{2} \) |
| 31 | \( 1 + 184 T + p^{3} T^{2} \) |
| 37 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 362 T + p^{3} T^{2} \) |
| 43 | \( 1 + 76 T + p^{3} T^{2} \) |
| 47 | \( 1 + 452 T + p^{3} T^{2} \) |
| 53 | \( 1 + 382 T + p^{3} T^{2} \) |
| 59 | \( 1 + 464 T + p^{3} T^{2} \) |
| 61 | \( 1 + 358 T + p^{3} T^{2} \) |
| 67 | \( 1 - 700 T + p^{3} T^{2} \) |
| 71 | \( 1 + 748 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1058 T + p^{3} T^{2} \) |
| 79 | \( 1 + 976 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1008 T + p^{3} T^{2} \) |
| 89 | \( 1 + 386 T + p^{3} T^{2} \) |
| 97 | \( 1 + 614 T + p^{3} T^{2} \) |
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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
−8.277128111182670127197665698516, −7.79470786174600877214805138615, −7.17912818826609614255064823452, −6.20215113234700980468648256545, −5.36678659725402263954052212756, −4.78544687286091458554004896539, −3.59468348153758132672877801260, −3.06410583350988850963273638139, −1.60893225071301858449973478687, −0.33245213530920080717427301061,
0.33245213530920080717427301061, 1.60893225071301858449973478687, 3.06410583350988850963273638139, 3.59468348153758132672877801260, 4.78544687286091458554004896539, 5.36678659725402263954052212756, 6.20215113234700980468648256545, 7.17912818826609614255064823452, 7.79470786174600877214805138615, 8.277128111182670127197665698516
