| L(s) = 1 | + 2·11-s + 13-s + 2·17-s + 8·19-s − 23-s − 9·29-s − 4·31-s − 6·37-s + 2·41-s − 9·43-s + 4·47-s − 7·49-s + 13·53-s − 6·59-s − 5·61-s − 6·67-s − 2·71-s − 8·73-s + 17·79-s − 6·83-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.603·11-s + 0.277·13-s + 0.485·17-s + 1.83·19-s − 0.208·23-s − 1.67·29-s − 0.718·31-s − 0.986·37-s + 0.312·41-s − 1.37·43-s + 0.583·47-s − 49-s + 1.78·53-s − 0.781·59-s − 0.640·61-s − 0.733·67-s − 0.237·71-s − 0.936·73-s + 1.91·79-s − 0.658·83-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.82799877459616, −15.01905439920012, −14.76416018809914, −14.06740983086796, −13.54486875171171, −13.21291018702059, −12.26427838149095, −12.01446637013573, −11.41414731553850, −10.88051916641812, −10.20228731352043, −9.586893662649213, −9.190038489228808, −8.596093807630910, −7.766112357247105, −7.387288421992541, −6.792200576292919, −5.985904921631295, −5.449332618730897, −4.959187293072127, −3.901849598054841, −3.560377892940751, −2.820540181919133, −1.744067136300335, −1.212178227902052, 0,
1.212178227902052, 1.744067136300335, 2.820540181919133, 3.560377892940751, 3.901849598054841, 4.959187293072127, 5.449332618730897, 5.985904921631295, 6.792200576292919, 7.387288421992541, 7.766112357247105, 8.596093807630910, 9.190038489228808, 9.586893662649213, 10.20228731352043, 10.88051916641812, 11.41414731553850, 12.01446637013573, 12.26427838149095, 13.21291018702059, 13.54486875171171, 14.06740983086796, 14.76416018809914, 15.01905439920012, 15.82799877459616
