| L(s) = 1 | + 7·3-s − 9·5-s − 2·7-s + 22·9-s − 11·11-s − 63·15-s − 38·17-s + 44·19-s − 14·21-s − 175·23-s − 44·25-s − 35·27-s + 264·29-s − 159·31-s − 77·33-s + 18·35-s + 173·37-s − 220·41-s − 542·43-s − 198·45-s + 264·47-s − 339·49-s − 266·51-s − 682·53-s + 99·55-s + 308·57-s + 421·59-s + ⋯ |
| L(s) = 1 | + 1.34·3-s − 0.804·5-s − 0.107·7-s + 0.814·9-s − 0.301·11-s − 1.08·15-s − 0.542·17-s + 0.531·19-s − 0.145·21-s − 1.58·23-s − 0.351·25-s − 0.249·27-s + 1.69·29-s − 0.921·31-s − 0.406·33-s + 0.0869·35-s + 0.768·37-s − 0.838·41-s − 1.92·43-s − 0.655·45-s + 0.819·47-s − 0.988·49-s − 0.730·51-s − 1.76·53-s + 0.242·55-s + 0.715·57-s + 0.928·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + p T \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 5 | \( 1 + 9 T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 13 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 + 38 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 + 175 T + p^{3} T^{2} \) |
| 29 | \( 1 - 264 T + p^{3} T^{2} \) |
| 31 | \( 1 + 159 T + p^{3} T^{2} \) |
| 37 | \( 1 - 173 T + p^{3} T^{2} \) |
| 41 | \( 1 + 220 T + p^{3} T^{2} \) |
| 43 | \( 1 + 542 T + p^{3} T^{2} \) |
| 47 | \( 1 - 264 T + p^{3} T^{2} \) |
| 53 | \( 1 + 682 T + p^{3} T^{2} \) |
| 59 | \( 1 - 421 T + p^{3} T^{2} \) |
| 61 | \( 1 + 308 T + p^{3} T^{2} \) |
| 67 | \( 1 - 177 T + p^{3} T^{2} \) |
| 71 | \( 1 + 365 T + p^{3} T^{2} \) |
| 73 | \( 1 + 528 T + p^{3} T^{2} \) |
| 79 | \( 1 + 686 T + p^{3} T^{2} \) |
| 83 | \( 1 - 698 T + p^{3} T^{2} \) |
| 89 | \( 1 - 967 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1127 T + p^{3} T^{2} \) |
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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
−9.548873876718196813924116081832, −8.505327869480137519909201791941, −8.079073383655733204490152937517, −7.30096549367514800771588857531, −6.17261944066877069330089061159, −4.74947758116751297214882709594, −3.77047190799024096186824042095, −2.98611723546088308152633628332, −1.84531309967112419989660036968, 0,
1.84531309967112419989660036968, 2.98611723546088308152633628332, 3.77047190799024096186824042095, 4.74947758116751297214882709594, 6.17261944066877069330089061159, 7.30096549367514800771588857531, 8.079073383655733204490152937517, 8.505327869480137519909201791941, 9.548873876718196813924116081832
