| L(s) = 1 | − 3-s − 2·5-s − 2·7-s + 9-s − 13-s + 2·15-s + 2·19-s + 2·21-s + 8·23-s − 25-s − 27-s + 4·29-s − 4·31-s + 4·35-s − 6·37-s + 39-s − 6·43-s − 2·45-s − 3·49-s − 2·53-s − 2·57-s + 12·59-s + 6·61-s − 2·63-s + 2·65-s − 8·67-s − 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.516·15-s + 0.458·19-s + 0.436·21-s + 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.742·29-s − 0.718·31-s + 0.676·35-s − 0.986·37-s + 0.160·39-s − 0.914·43-s − 0.298·45-s − 3/7·49-s − 0.274·53-s − 0.264·57-s + 1.56·59-s + 0.768·61-s − 0.251·63-s + 0.248·65-s − 0.977·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.43543294350233, −13.68377527362337, −13.17420769191652, −12.78356047612573, −12.27276489971791, −11.80759913159600, −11.32951548371355, −10.97209979302438, −10.22238550001719, −9.913588711222744, −9.288764713973074, −8.686837121928504, −8.252650428720573, −7.430711303654102, −7.141288900839416, −6.664403226712068, −6.064202799479870, −5.294775634403808, −4.981511064099950, −4.283838253367647, −3.558276474311687, −3.247031579681682, −2.465620627717732, −1.514926203662906, −0.7085263895346610, 0,
0.7085263895346610, 1.514926203662906, 2.465620627717732, 3.247031579681682, 3.558276474311687, 4.283838253367647, 4.981511064099950, 5.294775634403808, 6.064202799479870, 6.664403226712068, 7.141288900839416, 7.430711303654102, 8.252650428720573, 8.686837121928504, 9.288764713973074, 9.913588711222744, 10.22238550001719, 10.97209979302438, 11.32951548371355, 11.80759913159600, 12.27276489971791, 12.78356047612573, 13.17420769191652, 13.68377527362337, 14.43543294350233
