| L(s) = 1 | + 2·3-s + 5·5-s − 8·7-s − 23·9-s − 44·11-s + 10·15-s − 74·17-s − 19·19-s − 16·21-s − 84·23-s + 25·25-s − 100·27-s + 266·29-s − 136·31-s − 88·33-s − 40·35-s + 424·37-s + 470·41-s + 236·43-s − 115·45-s + 240·47-s − 279·49-s − 148·51-s + 36·53-s − 220·55-s − 38·57-s − 736·59-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 0.447·5-s − 0.431·7-s − 0.851·9-s − 1.20·11-s + 0.172·15-s − 1.05·17-s − 0.229·19-s − 0.166·21-s − 0.761·23-s + 1/5·25-s − 0.712·27-s + 1.70·29-s − 0.787·31-s − 0.464·33-s − 0.193·35-s + 1.88·37-s + 1.79·41-s + 0.836·43-s − 0.380·45-s + 0.744·47-s − 0.813·49-s − 0.406·51-s + 0.0933·53-s − 0.539·55-s − 0.0883·57-s − 1.62·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.669709648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.669709648\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 19 | \( 1 + p T \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 + 74 T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 - 266 T + p^{3} T^{2} \) |
| 31 | \( 1 + 136 T + p^{3} T^{2} \) |
| 37 | \( 1 - 424 T + p^{3} T^{2} \) |
| 41 | \( 1 - 470 T + p^{3} T^{2} \) |
| 43 | \( 1 - 236 T + p^{3} T^{2} \) |
| 47 | \( 1 - 240 T + p^{3} T^{2} \) |
| 53 | \( 1 - 36 T + p^{3} T^{2} \) |
| 59 | \( 1 + 736 T + p^{3} T^{2} \) |
| 61 | \( 1 - 650 T + p^{3} T^{2} \) |
| 67 | \( 1 - 830 T + p^{3} T^{2} \) |
| 71 | \( 1 - 216 T + p^{3} T^{2} \) |
| 73 | \( 1 - 254 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1220 T + p^{3} T^{2} \) |
| 83 | \( 1 - 688 T + p^{3} T^{2} \) |
| 89 | \( 1 - 102 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1280 T + p^{3} T^{2} \) |
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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
−9.175939188869187035088638039043, −8.249961694339215188202467643657, −7.72365534701057035445487943902, −6.51101100088812268525297563017, −5.92275098028355916319992649787, −4.99170067614985256123450072509, −3.97236402501917678855594525271, −2.68488309706221915600658661881, −2.35211495835571456988403663253, −0.58308632200016632166334253004,
0.58308632200016632166334253004, 2.35211495835571456988403663253, 2.68488309706221915600658661881, 3.97236402501917678855594525271, 4.99170067614985256123450072509, 5.92275098028355916319992649787, 6.51101100088812268525297563017, 7.72365534701057035445487943902, 8.249961694339215188202467643657, 9.175939188869187035088638039043
