| L(s) = 1 | − 3·3-s + 5.36·5-s − 7·7-s + 9·9-s + 11.0·11-s − 1.61·13-s − 16.0·15-s + 126.·17-s + 133.·19-s + 21·21-s + 23·23-s − 96.2·25-s − 27·27-s − 259.·29-s + 138.·31-s − 33.2·33-s − 37.5·35-s − 22.5·37-s + 4.84·39-s + 411.·41-s − 267.·43-s + 48.2·45-s + 297.·47-s + 49·49-s − 380.·51-s − 1.19·53-s + 59.4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.479·5-s − 0.377·7-s + 0.333·9-s + 0.303·11-s − 0.0344·13-s − 0.277·15-s + 1.80·17-s + 1.61·19-s + 0.218·21-s + 0.208·23-s − 0.769·25-s − 0.192·27-s − 1.66·29-s + 0.801·31-s − 0.175·33-s − 0.181·35-s − 0.100·37-s + 0.0198·39-s + 1.56·41-s − 0.949·43-s + 0.159·45-s + 0.924·47-s + 0.142·49-s − 1.04·51-s − 0.00309·53-s + 0.145·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.087204803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.087204803\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 5.36T + 125T^{2} \) |
| 11 | \( 1 - 11.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 1.61T + 2.19e3T^{2} \) |
| 17 | \( 1 - 126.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 133.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 259.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 138.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 22.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 411.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 267.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 297.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.19T + 1.48e5T^{2} \) |
| 59 | \( 1 + 821.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 69.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 971.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 137.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.61T + 3.89e5T^{2} \) |
| 79 | \( 1 - 701.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 960.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.46e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.03e3T + 9.12e5T^{2} \) |
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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
−9.159014428390580108031817932954, −7.70447458581547772120153057764, −7.45836239841458004934193390478, −6.21348449132329912865611536528, −5.72843880341163846799525886909, −5.00873083959125141653134102494, −3.79703079770736989325972121742, −3.01324323844802323993157424305, −1.64061217377155985943871847326, −0.72802494638702395556745454169,
0.72802494638702395556745454169, 1.64061217377155985943871847326, 3.01324323844802323993157424305, 3.79703079770736989325972121742, 5.00873083959125141653134102494, 5.72843880341163846799525886909, 6.21348449132329912865611536528, 7.45836239841458004934193390478, 7.70447458581547772120153057764, 9.159014428390580108031817932954
