L(s) = 1 | + 1.41·5-s − 3·9-s − 7.07·13-s − 8·17-s − 2.99·25-s + 4.24·29-s + 9.89·37-s − 8·41-s − 4.24·45-s − 7·49-s + 12.7·53-s − 15.5·61-s − 10.0·65-s + 6·73-s + 9·81-s − 11.3·85-s − 10·89-s − 8·97-s − 12.7·101-s + 18.3·109-s + 14·113-s + 21.2·117-s + ⋯ |
L(s) = 1 | + 0.632·5-s − 9-s − 1.96·13-s − 1.94·17-s − 0.599·25-s + 0.787·29-s + 1.62·37-s − 1.24·41-s − 0.632·45-s − 49-s + 1.74·53-s − 1.99·61-s − 1.24·65-s + 0.702·73-s + 81-s − 1.22·85-s − 1.05·89-s − 0.812·97-s − 1.26·101-s + 1.76·109-s + 1.31·113-s + 1.96·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 7.07T + 13T^{2} \) |
| 17 | \( 1 + 8T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.534483589371659383704577534942, −8.816487043176947810052863488761, −7.907819243526066736912848950458, −6.90704385413515130409907026623, −6.14044438985011453938621196450, −5.15292612052541076462457869194, −4.39277121804973977094034099524, −2.80084148899375299910265298036, −2.14155945791372166988925316299, 0,
2.14155945791372166988925316299, 2.80084148899375299910265298036, 4.39277121804973977094034099524, 5.15292612052541076462457869194, 6.14044438985011453938621196450, 6.90704385413515130409907026623, 7.907819243526066736912848950458, 8.816487043176947810052863488761, 9.534483589371659383704577534942