| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 3·11-s − 5·13-s + 14-s + 16-s − 6·19-s − 20-s + 3·22-s + 8·23-s − 4·25-s + 5·26-s − 28-s − 4·29-s + 6·31-s − 32-s + 35-s + 5·37-s + 6·38-s + 40-s + 4·41-s + 43-s − 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.904·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s − 1.37·19-s − 0.223·20-s + 0.639·22-s + 1.66·23-s − 4/5·25-s + 0.980·26-s − 0.188·28-s − 0.742·29-s + 1.07·31-s − 0.176·32-s + 0.169·35-s + 0.821·37-s + 0.973·38-s + 0.158·40-s + 0.624·41-s + 0.152·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−15.35340882918496, −14.69355192125405, −14.40977654670893, −13.33169424652546, −13.03290187895800, −12.55258089318310, −11.97494087023284, −11.34192037192249, −10.93546897792914, −10.28692457649320, −9.889537145217668, −9.314569867929235, −8.759762852467180, −8.092704575761071, −7.696640362463503, −7.093562512403296, −6.672291331786579, −5.851405280824438, −5.288224170983051, −4.527246709857984, −4.018353972730278, −2.911392211475277, −2.653043691558248, −1.867316944231998, −0.7114927316963196, 0,
0.7114927316963196, 1.867316944231998, 2.653043691558248, 2.911392211475277, 4.018353972730278, 4.527246709857984, 5.288224170983051, 5.851405280824438, 6.672291331786579, 7.093562512403296, 7.696640362463503, 8.092704575761071, 8.759762852467180, 9.314569867929235, 9.889537145217668, 10.28692457649320, 10.93546897792914, 11.34192037192249, 11.97494087023284, 12.55258089318310, 13.03290187895800, 13.33169424652546, 14.40977654670893, 14.69355192125405, 15.35340882918496
