| L(s) = 1 | − 3·3-s + 5·5-s + 9·9-s + 11·11-s − 42·13-s − 15·15-s − 14·17-s − 52·19-s + 96·23-s + 25·25-s − 27·27-s − 26·29-s − 144·31-s − 33·33-s + 126·37-s + 126·39-s + 58·41-s + 364·43-s + 45·45-s − 328·47-s − 343·49-s + 42·51-s − 50·53-s + 55·55-s + 156·57-s − 284·59-s − 794·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s − 0.896·13-s − 0.258·15-s − 0.199·17-s − 0.627·19-s + 0.870·23-s + 1/5·25-s − 0.192·27-s − 0.166·29-s − 0.834·31-s − 0.174·33-s + 0.559·37-s + 0.517·39-s + 0.220·41-s + 1.29·43-s + 0.149·45-s − 1.01·47-s − 49-s + 0.115·51-s − 0.129·53-s + 0.134·55-s + 0.362·57-s − 0.626·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 11 | \( 1 - p T \) |
| good | 7 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 26 T + p^{3} T^{2} \) |
| 31 | \( 1 + 144 T + p^{3} T^{2} \) |
| 37 | \( 1 - 126 T + p^{3} T^{2} \) |
| 41 | \( 1 - 58 T + p^{3} T^{2} \) |
| 43 | \( 1 - 364 T + p^{3} T^{2} \) |
| 47 | \( 1 + 328 T + p^{3} T^{2} \) |
| 53 | \( 1 + 50 T + p^{3} T^{2} \) |
| 59 | \( 1 + 284 T + p^{3} T^{2} \) |
| 61 | \( 1 + 794 T + p^{3} T^{2} \) |
| 67 | \( 1 + 316 T + p^{3} T^{2} \) |
| 71 | \( 1 + 280 T + p^{3} T^{2} \) |
| 73 | \( 1 + 358 T + p^{3} T^{2} \) |
| 79 | \( 1 - 784 T + p^{3} T^{2} \) |
| 83 | \( 1 - 324 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1398 T + p^{3} T^{2} \) |
| 97 | \( 1 + 894 T + p^{3} T^{2} \) |
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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
−9.656905577971047262151725360981, −9.068774202851517893219769020188, −7.82584260910518473904086008208, −6.92066874403063263069664478320, −6.09708839549662697975574164854, −5.13833631927231244764585446237, −4.25653746599888616178700595871, −2.80019645083200556978194164029, −1.52028883848851401307599578277, 0,
1.52028883848851401307599578277, 2.80019645083200556978194164029, 4.25653746599888616178700595871, 5.13833631927231244764585446237, 6.09708839549662697975574164854, 6.92066874403063263069664478320, 7.82584260910518473904086008208, 9.068774202851517893219769020188, 9.656905577971047262151725360981
