| L(s) = 1 | + 5.65·3-s − 1.41·5-s + 8·7-s + 5.00·9-s + 5.65·11-s − 4.24·13-s − 8.00·15-s − 24·17-s − 130.·19-s + 45.2·21-s − 184·23-s − 123·25-s − 124.·27-s + 97.5·29-s + 176·31-s + 32.0·33-s − 11.3·35-s − 304.·37-s − 24·39-s + 200·41-s − 141.·43-s − 7.07·45-s + 208·47-s − 279·49-s − 135.·51-s + 371.·53-s − 8.00·55-s + ⋯ |
| L(s) = 1 | + 1.08·3-s − 0.126·5-s + 0.431·7-s + 0.185·9-s + 0.155·11-s − 0.0905·13-s − 0.137·15-s − 0.342·17-s − 1.57·19-s + 0.470·21-s − 1.66·23-s − 0.983·25-s − 0.887·27-s + 0.624·29-s + 1.01·31-s + 0.168·33-s − 0.0546·35-s − 1.35·37-s − 0.0985·39-s + 0.761·41-s − 0.501·43-s − 0.0234·45-s + 0.645·47-s − 0.813·49-s − 0.372·51-s + 0.963·53-s − 0.0196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 5.65T + 27T^{2} \) |
| 5 | \( 1 + 1.41T + 125T^{2} \) |
| 7 | \( 1 - 8T + 343T^{2} \) |
| 11 | \( 1 - 5.65T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.24T + 2.19e3T^{2} \) |
| 17 | \( 1 + 24T + 4.91e3T^{2} \) |
| 19 | \( 1 + 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 184T + 1.21e4T^{2} \) |
| 29 | \( 1 - 97.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 176T + 2.97e4T^{2} \) |
| 37 | \( 1 + 304.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 200T + 6.89e4T^{2} \) |
| 43 | \( 1 + 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 208T + 1.03e5T^{2} \) |
| 53 | \( 1 - 371.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 152.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 479.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 684.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 936T + 3.57e5T^{2} \) |
| 73 | \( 1 - 406T + 3.89e5T^{2} \) |
| 79 | \( 1 - 768T + 4.93e5T^{2} \) |
| 83 | \( 1 - 514.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 472T + 9.12e5T^{2} \) |
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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
−8.934776165870403989158366856290, −8.319685161738985748786203386338, −7.81374097265251670950420156924, −6.67034939455873319306838204801, −5.77885154829350627626257964672, −4.45360437916854183200679633050, −3.78696793636416160077079014918, −2.55361834220837218316019189523, −1.80198053178166527907640796595, 0,
1.80198053178166527907640796595, 2.55361834220837218316019189523, 3.78696793636416160077079014918, 4.45360437916854183200679633050, 5.77885154829350627626257964672, 6.67034939455873319306838204801, 7.81374097265251670950420156924, 8.319685161738985748786203386338, 8.934776165870403989158366856290
